.. _fmpz-mat:

**fmpz_mat.h** -- matrices over the integers
===============================================================================

The :type:`fmpz_mat_t` data type represents dense matrices of
multiprecision integers, implemented using :type:`fmpz` vectors.

No automatic resizing is performed: in general, the user must provide
matrices of correct dimensions for both input and output variables.
Output variables are *not* allowed to be aliased with input variables
unless otherwise noted.

Matrices are indexed from zero: an `m \times n` matrix has rows of
index `0,1,\ldots,m-1` and columns of index `0,1,\ldots,n-1`. One or
both of `m` and `n` may be zero.

Elements of a matrix can be read or written using the
``fmpz_mat_entry`` macro, which returns a reference to the entry at a
given row and column index. This reference can be passed as an input
or output :type:`fmpz_t` variable to any function in the ``fmpz``
module for direct manipulation.

Simple example
--------------

The following example creates the `2 \times 2` matrix `A` with value
`2i+j` at row `i` and column `j`, computes `B = A^2`, and prints both
matrices.

.. code:: c

   #include "fmpz.h"
   #include "fmpz_mat.h"

   int main()
   {
       long i, j;
       fmpz_mat_t A;
       fmpz_mat_t B;
       fmpz_mat_init(A, 2, 2);
       fmpz_mat_init(B, 2, 2);
       for (i = 0; i < 2; i++)
           for (j = 0; j < 2; j++)
               fmpz_set_ui(fmpz_mat_entry(A, i, j), 2*i+j);
       fmpz_mat_mul(B, A, A);
       flint_printf("A = \n");
       fmpz_mat_print_pretty(A);
       flint_printf("A^2 = \n");
       fmpz_mat_print_pretty(B);
       fmpz_mat_clear(A);
       fmpz_mat_clear(B);
   }

The output is:

::

   A =
   [[0 1]
   [2 3]]
   A^2 =
   [[2 3]
   [6 11]]

Types, macros and constants
-------------------------------------------------------------------------------

.. type:: fmpz_mat_struct

.. type:: fmpz_mat_t

Memory management
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_init(fmpz_mat_t mat, slong rows, slong cols)

    Initialises a matrix with the given number of rows and columns for use.

.. function:: void fmpz_mat_clear(fmpz_mat_t mat)

    Clears the given matrix.


Basic assignment and manipulation
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_set(fmpz_mat_t mat1, const fmpz_mat_t mat2)

    Sets ``mat1`` to a copy of ``mat2``. The dimensions of
    ``mat1`` and ``mat2`` must be the same.

.. function:: void fmpz_mat_init_set(fmpz_mat_t mat, const fmpz_mat_t src)

    Initialises the matrix ``mat`` to the same size as ``src`` and
    sets it to a copy of ``src``.

.. function:: slong fmpz_mat_nrows(const fmpz_mat_t mat)
              slong fmpz_mat_ncols(const fmpz_mat_t mat)

    Returns respectively the number of rows and columns of the matrix.

.. function:: void fmpz_mat_swap(fmpz_mat_t mat1, fmpz_mat_t mat2)

    Swaps two matrices. The dimensions of ``mat1`` and ``mat2``
    are allowed to be different.

.. function:: void fmpz_mat_swap_entrywise(fmpz_mat_t mat1, fmpz_mat_t mat2)

    Swaps two matrices by swapping the individual entries rather than swapping
    the contents of the structs.

.. function:: fmpz * fmpz_mat_entry(const fmpz_mat_t mat, slong i, slong j)

    Returns a reference to the entry of ``mat`` at row `i` and column `j`.
    This reference can be passed as an input or output variable to any
    function in the ``fmpz`` module for direct manipulation.

    Both `i` and `j` must not exceed the dimensions of the matrix.

    This function is implemented as a macro.

.. function:: void fmpz_mat_zero(fmpz_mat_t mat)

    Sets all entries of ``mat`` to 0.

.. function:: void fmpz_mat_one(fmpz_mat_t mat)

    Sets ``mat`` to the unit matrix, having ones on the main diagonal
    and zeroes elsewhere. If ``mat`` is nonsquare, it is set to the
    truncation of a unit matrix.

.. function:: void fmpz_mat_swap_rows(fmpz_mat_t mat, slong * perm, slong r, slong s)

    Swaps rows ``r`` and ``s`` of ``mat``.  If ``perm`` is non-``NULL``, the
    permutation of the rows will also be applied to ``perm``.

.. function:: void fmpz_mat_swap_cols(fmpz_mat_t mat, slong * perm, slong r, slong s)

    Swaps columns ``r`` and ``s`` of ``mat``.  If ``perm`` is non-``NULL``, the
    permutation of the columns will also be applied to ``perm``.

.. function:: void fmpz_mat_invert_rows(fmpz_mat_t mat, slong * perm)

    Swaps rows ``i`` and ``r - i`` of ``mat`` for ``0 <= i < r/2``, where
    ``r`` is the number of rows of ``mat``. If ``perm`` is non-``NULL``, the
    permutation of the rows will also be applied to ``perm``.

.. function:: void fmpz_mat_invert_cols(fmpz_mat_t mat, slong * perm)

    Swaps columns ``i`` and ``c - i`` of ``mat`` for ``0 <= i < c/2``, where
    ``c`` is the number of columns of ``mat``. If ``perm`` is non-``NULL``, the
    permutation of the columns will also be applied to ``perm``.

Window
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_window_init(fmpz_mat_t window, const fmpz_mat_t mat, slong r1, slong c1, slong r2, slong c2)

    Initializes the matrix ``window`` to be an ``r2 - r1`` by
    ``c2 - c1`` submatrix of ``mat`` whose ``(0,0)`` entry
    is the ``(r1, c1)`` entry of ``mat``. The memory for the
    elements of ``window`` is shared with ``mat``.

.. function:: void fmpz_mat_window_clear(fmpz_mat_t window)

    Clears the matrix ``window`` and releases any memory that it
    uses. Note that the memory to the underlying matrix that
    ``window`` points to is not freed.


Random matrix generation
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_randbits(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)

    Sets the entries of ``mat`` to random signed integers whose absolute
    values have the given number of binary bits.

.. function:: void fmpz_mat_randtest(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)

    Sets the entries of ``mat`` to random signed integers whose
    absolute values have a random number of bits up to the given number
    of bits inclusive.

.. function:: void fmpz_mat_randintrel(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)

    Sets ``mat`` to be a random *integer relations* matrix, with
    signed entries up to the given number of bits.

    The number of columns of ``mat`` must be equal to one more than
    the number of rows. The format of the matrix is a set of random integers
    in the left hand column and an identity matrix in the remaining square
    submatrix.

.. function:: void fmpz_mat_randsimdioph(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits, flint_bitcnt_t bits2)

    Sets ``mat`` to a random *simultaneous diophantine* matrix.

    The matrix must be square. The top left entry is set to ``2^bits2``.
    The remainder of that row is then set to signed random integers of the
    given number of binary bits. The remainder of the first column is zero.
    Running down the rest of the diagonal are the values ``2^bits`` with
    all remaining entries zero.

.. function:: void fmpz_mat_randntrulike(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits, ulong q)

    Sets a square matrix ``mat`` of even dimension to a random
    *NTRU like* matrix.

    The matrix is broken into four square submatrices. The top left submatrix
    is set to the identity. The bottom left submatrix is set to the zero
    matrix. The bottom right submatrix is set to `q` times the identity matrix.
    Finally the top right submatrix has the following format. A random vector
    `h` of length `r/2` is created, with random signed entries of the given
    number of bits. Then entry `(i, j)` of the submatrix is set to
    `h[i + j \bmod{r/2}]`.

.. function:: void fmpz_mat_randntrulike2(fmpz_mat_t mat, flint_rand_t state, flint_bitcnt_t bits, ulong q)

    Sets a square matrix ``mat`` of even dimension to a random
    *NTRU like* matrix.

    The matrix is broken into four square submatrices. The top left submatrix
    is set to `q` times the identity matrix. The top right submatrix is set to
    the zero matrix. The bottom right submatrix is set to the identity matrix.
    Finally the bottom left submatrix has the following format. A random vector
    `h` of length `r/2` is created, with random signed entries of the given
    number of bits. Then entry `(i, j)` of the submatrix is set to
    `h[i + j \bmod{r/2}]`.

.. function:: void fmpz_mat_randajtai(fmpz_mat_t mat, flint_rand_t state, double alpha)

    Sets a square matrix ``mat`` to a random *ajtai* matrix.
    The diagonal entries `(i, i)` are set to a random entry in the range
    `[1, 2^{b-1}]` inclusive where `b = \lfloor(2 r - i)^\alpha\rfloor` for some
    double parameter `\alpha`. The entries below the diagonal in column `i`
    are set to a random entry in the range `(-2^b + 1, 2^b - 1)` whilst the
    entries to the right of the diagonal in row `i` are set to zero.

.. function:: int fmpz_mat_randpermdiag(fmpz_mat_t mat, flint_rand_t state, const fmpz * diag, slong n)

    Sets ``mat`` to a random permutation of the rows and columns of a
    given diagonal matrix. The diagonal matrix is specified in the form of
    an array of the `n` initial entries on the main diagonal.

    The return value is `0` or `1` depending on whether the permutation is
    even or odd.

.. function:: void fmpz_mat_randrank(fmpz_mat_t mat, flint_rand_t state, slong rank, flint_bitcnt_t bits)

    Sets ``mat`` to a random sparse matrix with the given rank,
    having exactly as many non-zero elements as the rank, with the
    nonzero elements being random integers of the given bit size.

    The matrix can be transformed into a dense matrix with unchanged
    rank by subsequently calling :func:`fmpz_mat_randops`.

.. function:: void fmpz_mat_randdet(fmpz_mat_t mat, flint_rand_t state, const fmpz_t det)

    Sets ``mat`` to a random sparse matrix with minimal number of
    nonzero entries such that its determinant has the given value.

    Note that the matrix will be zero if ``det`` is zero.
    In order to generate a non-zero singular matrix, the function
    :func:`fmpz_mat_randrank` can be used.

    The matrix can be transformed into a dense matrix with unchanged
    determinant by subsequently calling :func:`fmpz_mat_randops`.

.. function:: void fmpz_mat_randops(fmpz_mat_t mat, flint_rand_t state, slong count)

    Randomises ``mat`` by performing elementary row or column operations.
    More precisely, at most ``count`` random additions or subtractions of
    distinct rows and columns will be performed. This leaves the rank
    (and for square matrices, the determinant) unchanged.



Input and output
--------------------------------------------------------------------------------


.. function:: int fmpz_mat_fprint(FILE * file, const fmpz_mat_t mat)

    Prints the given matrix to the stream ``file``.  The format is
    the number of rows, a space, the number of columns, two spaces, then
    a space separated list of coefficients, one row after the other.

    In case of success, returns a positive value;  otherwise, returns
    a non-positive value.

.. function:: int fmpz_mat_fprint_pretty(FILE * file, const fmpz_mat_t mat)

    Prints the given matrix to the stream ``file``.  The format is an
    opening square bracket, then on each line a row of the matrix, followed
    by a closing square bracket. Each row is written as an opening square
    bracket followed by a space separated list of coefficients followed
    by a closing square bracket.

    In case of success, returns a positive value;  otherwise, returns
    a non-positive value.

.. function:: int fmpz_mat_print(const fmpz_mat_t mat)

    Prints the given matrix to the stream ``stdout``.  For further
    details, see :func:`fmpz_mat_fprint`.

.. function:: int fmpz_mat_print_pretty(const fmpz_mat_t mat)

    Prints the given matrix to ``stdout``.  For further details,
    see :func:`fmpz_mat_fprint_pretty`.

.. function:: int fmpz_mat_fread(FILE* file, fmpz_mat_t mat)

    Reads a matrix from the stream ``file``, storing the result
    in ``mat``.  The expected format is the number of rows, a
    space, the number of columns, two spaces, then a space separated
    list of coefficients, one row after the other.

    In case of success, returns a positive number.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_mat_read(fmpz_mat_t mat)

    Reads a matrix from ``stdin``, storing the result
    in ``mat``.

    In case of success, returns a positive number.  In case of failure,
    returns a non-positive value.


Comparison
--------------------------------------------------------------------------------


.. function:: int fmpz_mat_equal(const fmpz_mat_t mat1, const fmpz_mat_t mat2)

    Returns a non-zero value if ``mat1`` and ``mat2`` have
    the same dimensions and entries, and zero otherwise.

.. function:: int fmpz_mat_is_zero(const fmpz_mat_t mat)

    Returns a non-zero value if all entries ``mat`` are zero, and
    otherwise returns zero.

.. function:: int fmpz_mat_is_one(const fmpz_mat_t mat)

    Returns a non-zero value if ``mat`` is the unit matrix or the truncation
    of a unit matrix, and otherwise returns zero.

.. function:: int fmpz_mat_is_empty(const fmpz_mat_t mat)

    Returns a non-zero value if the number of rows or the number of
    columns in ``mat`` is zero, and otherwise returns
    zero.

.. function:: int fmpz_mat_is_square(const fmpz_mat_t mat)

    Returns a non-zero value if the number of rows is equal to the
    number of columns in ``mat``, and otherwise returns zero.

.. function:: int fmpz_mat_is_zero_row(const fmpz_mat_t mat, slong i)

    Returns a non-zero value if row `i` of ``mat`` is zero.

.. function:: int fmpz_mat_equal_col(fmpz_mat_t M, slong m, slong n)

    Returns `1` if columns `m` and `n` of the matrix `M` are equal, otherwise
    returns `0`.

.. function:: int fmpz_mat_equal_row(fmpz_mat_t M, slong m, slong n)

    Returns `1` if rows `m` and `n` of the matrix `M` are equal, otherwise
    returns `0`.


Transpose
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_transpose(fmpz_mat_t B, const fmpz_mat_t A)

    Sets `B` to `A^T`, the transpose of `A`. Dimensions must be compatible.
    `A` and `B` are allowed to be the same object if `A` is a square matrix.



Concatenate
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_concat_vertical(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_mat_t mat2)

    Sets ``res`` to vertical concatenation of (``mat1``, ``mat2``)
    in that order. Matrix dimensions: ``mat1``: `m \times n`,
    ``mat2``: `k \times n`, ``res``: `(m + k) \times n`.

.. function:: void fmpz_mat_concat_horizontal(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_mat_t mat2)

    Sets ``res`` to horizontal concatenation of (``mat1``, ``mat2``)
    in that order. Matrix dimensions: ``mat1``: `m \times n`,
    ``mat2``: `m \times k`, ``res``: `m \times (n + k)`.


Modular reduction and reconstruction
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_get_nmod_mat(nmod_mat_t Amod, const fmpz_mat_t A)

    Sets the entries of ``Amod`` to the entries of ``A`` reduced
    by the modulus of ``Amod``.

.. function:: void fmpz_mat_set_nmod_mat(fmpz_mat_t A, const nmod_mat_t Amod)

    Sets the entries of ``Amod`` to the residues in ``Amod``,
    normalised to the interval `-m/2 <= r < m/2` where `m` is the modulus.

.. function:: void fmpz_mat_set_nmod_mat_unsigned(fmpz_mat_t A, const nmod_mat_t Amod)

    Sets the entries of ``Amod`` to the residues in ``Amod``,
    normalised to the interval `0 <= r < m` where `m` is the modulus.

.. function:: void fmpz_mat_CRT_ui(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_t m1, const nmod_mat_t mat2, int sign)

    Given ``mat1`` with entries modulo ``m`` and ``mat2``
    with modulus `n`, sets ``res`` to the CRT reconstruction modulo `mn`
    with entries satisfying `-mn/2 <= c < mn/2` (if sign = 1)
    or `0 <= c < mn` (if sign = 0).

.. function:: void fmpz_mat_multi_mod_ui_precomp(nmod_mat_t * residues, slong nres, const fmpz_mat_t mat, const fmpz_comb_t comb, fmpz_comb_temp_t temp)

    Sets each of the ``nres`` matrices in ``residues`` to ``mat`` reduced modulo
    the modulus of the respective matrix, given precomputed ``comb`` and
    ``comb_temp`` structures.

    Note: ``fmpz.h`` must be included **before** ``fmpz_mat.h`` in order for
    this function to be declared.

.. function:: void fmpz_mat_multi_mod_ui(nmod_mat_t * residues, slong nres, const fmpz_mat_t mat)

    Sets each of the ``nres`` matrices in ``residues`` to ``mat``
    reduced modulo the modulus of the respective matrix.

    This function is provided for convenience purposes.
    For reducing or reconstructing multiple integer matrices over the same
    set of moduli, it is faster to use ``fmpz_mat_multi_mod_precomp``.

.. function:: void fmpz_mat_multi_CRT_ui_precomp(fmpz_mat_t mat, nmod_mat_t * const residues, slong nres, const fmpz_comb_t comb, fmpz_comb_temp_t temp, int sign)

    Reconstructs ``mat`` from its images modulo the ``nres`` matrices in
    ``residues``, given precomputed ``comb`` and ``comb_temp`` structures.

    Note: ``fmpz.h`` must be included **before** ``fmpz_mat.h`` in order for
    this function to be declared.

.. function:: void fmpz_mat_multi_CRT_ui(fmpz_mat_t mat, nmod_mat_t * const residues, slong nres, int sign)

    Reconstructs ``mat`` from its images modulo the ``nres`` matrices
    in ``residues``.

    This function is provided for convenience purposes.
    For reducing or reconstructing multiple integer matrices over the same
    set of moduli, it is faster to use :func:`fmpz_mat_multi_CRT_ui_precomp`.


Addition and subtraction
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_add(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the elementwise sum `A + B`. All inputs must
    be of the same size. Aliasing is allowed.

.. function:: void fmpz_mat_sub(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the elementwise difference `A - B`. All inputs must
    be of the same size. Aliasing is allowed.

.. function:: void fmpz_mat_neg(fmpz_mat_t B, const fmpz_mat_t A)

    Sets ``B`` to the elementwise negation of ``A``. Both inputs
    must be of the same size. Aliasing is allowed.


Matrix-scalar arithmetic
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_scalar_mul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)
              void fmpz_mat_scalar_mul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)
              void fmpz_mat_scalar_mul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)

    Set ``B = A*c`` where ``A`` is an ``fmpz_mat_t`` and ``c``
    is a scalar respectively of type ``slong``, ``ulong``,
    or ``fmpz_t``. The dimensions of ``A`` and ``B`` must
    be compatible.

.. function:: void fmpz_mat_scalar_addmul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)
              void fmpz_mat_scalar_addmul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)
              void fmpz_mat_scalar_addmul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)

    Set ``B = B + A*c`` where ``A`` is an ``fmpz_mat_t`` and ``c``
    is a scalar respectively of type ``slong``, ``ulong``,
    or ``fmpz_t``. The dimensions of ``A`` and ``B`` must
    be compatible.

.. function:: void fmpz_mat_scalar_submul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)
              void fmpz_mat_scalar_submul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)
              void fmpz_mat_scalar_submul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)

    Set ``B = B - A*c`` where ``A`` is an ``fmpz_mat_t`` and ``c``
    is a scalar respectively of type ``slong``, ``ulong``,
    or ``fmpz_t``. The dimensions of ``A`` and ``B`` must
    be compatible.

.. function:: void fmpz_mat_scalar_addmul_nmod_mat_ui(fmpz_mat_t B, const nmod_mat_t A, ulong c)
              void fmpz_mat_scalar_addmul_nmod_mat_fmpz(fmpz_mat_t B, const nmod_mat_t A, const fmpz_t c)

    Set ``B = B + A*c`` where ``A`` is an ``nmod_mat_t`` and ``c``
    is a scalar respectively of type ``ulong`` or ``fmpz_t``.
    The dimensions of ``A`` and ``B`` must be compatible.

.. function:: void fmpz_mat_scalar_divexact_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)
              void fmpz_mat_scalar_divexact_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)
              void fmpz_mat_scalar_divexact_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)

    Set ``A = B / c``, where ``B`` is an ``fmpz_mat_t`` and ``c``
    is a scalar respectively of type ``slong``, ``ulong``,
    or ``fmpz_t``, which is assumed to divide all elements of
    ``B`` exactly.

.. function:: void fmpz_mat_scalar_mul_2exp(fmpz_mat_t B, const fmpz_mat_t A, ulong exp)

    Set the matrix ``B`` to the matrix ``A``, of the same dimensions,
    multiplied by `2^{exp}`.

.. function:: void fmpz_mat_scalar_tdiv_q_2exp(fmpz_mat_t B, const fmpz_mat_t A, ulong exp)

    Set the matrix ``B`` to the matrix ``A``, of the same dimensions,
    divided by `2^{exp}`, rounding down towards zero.

.. function:: void fmpz_mat_scalar_smod(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t P)

    Set the matrix ``B`` to the matrix ``A``, of the same dimensions,
    with each entry reduced modulo `P` in the symmetric moduli system. We
    require `P > 0`.


Matrix multiplication
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_mul(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the matrix product `C = A B`. The matrices must have
    compatible dimensions for matrix multiplication. Aliasing
    is allowed.

    This function automatically switches between classical and
    multimodular multiplication, based on a heuristic comparison of
    the dimensions and entry sizes.

.. function:: void fmpz_mat_mul_classical(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the matrix product `C = A B` computed using
    classical matrix algorithm.

    The matrices must have compatible dimensions for matrix multiplication.
    No aliasing is allowed.

.. function:: void fmpz_mat_mul_strassen(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets `C = AB`. Dimensions must be compatible for matrix multiplication.
    `C` is not allowed to be aliased with `A` or `B`. Uses Strassen
    multiplication (the Strassen-Winograd variant).

.. function:: void _fmpz_mat_mul_multi_mod(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B, int sign, flint_bitcnt_t bits)
              void fmpz_mat_mul_multi_mod(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the matrix product `C = AB` computed using a multimodular
    algorithm. `C` is computed modulo several small prime numbers
    and reconstructed using the Chinese Remainder Theorem. This generally
    becomes more efficient than classical multiplication for large matrices.

    The absolute value of the elements of `C` should be `< 2^{\text{bits}}`,
    and ``sign`` should be `0` if the entries of `C` are known to be nonnegative
    and `1` otherwise. The function
    :func:`fmpz_mat_mul_multi_mod` calculates a rigorous bound automatically.
    If the default bound is too pessimistic, :func:`_fmpz_mat_mul_multi_mod`
    can be used with a custom bound.

    The matrices must have compatible dimensions for matrix multiplication.
    No aliasing is allowed.

.. function:: int fmpz_mat_mul_blas(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Tries to set `C = AB` using BLAS and returns `1` for success and `0` for failure.
    Dimensions must be compatible for matrix multiplication. No aliasing is allowed.
    This function currently will fail if the matrices are empty, their dimensions are too large, or their max bits size is over one million bits.

.. function:: void fmpz_mat_mul_fft(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Aliasing is allowed.

.. function:: void fmpz_mat_sqr(fmpz_mat_t B, const fmpz_mat_t A)

    Sets ``B`` to the square of the matrix ``A``, which must be
    a square matrix. Aliasing is allowed.
    The function calls :func:`fmpz_mat_mul` for dimensions less than 12 and
    calls :func:`fmpz_mat_sqr_bodrato` for cases in which the latter is faster.

.. function:: void fmpz_mat_sqr_bodrato(fmpz_mat_t B, const fmpz_mat_t A)

    Sets ``B`` to the square of the matrix ``A``, which must be
    a square matrix. Aliasing is allowed.
    The Bodrato algorithm is described in [Bodrato2010]_.
    It is highly efficient for squaring matrices which satisfy both the
    following conditions: (a) large elements,  (b) dimensions less than 150.


.. function:: void fmpz_mat_pow(fmpz_mat_t B, const fmpz_mat_t A, ulong e)

    Sets ``B`` to the matrix ``A`` raised to the power ``e``,
    where ``A`` must be a square matrix. Aliasing is allowed.


.. function:: void _fmpz_mat_mul_small(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    This internal function sets `C` to the matrix product `C = A B` computed
    using classical matrix algorithm assuming that all entries of `A` and `B`
    are small, that is, have bits `\le FLINT\_BITS - 2`. No aliasing is allowed.

.. function:: void _fmpz_mat_mul_double_word(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    This function is only for internal use and assumes that either:
        - the entries of `A` and `B` are all nonnegative and strictly less than `2^{2*FLINT\_BITS}`, or
        - the entries of `A` and `B` are all strictly less than `2^{2*FLINT\_BITS - 1}` in absolute value.

.. function:: void fmpz_mat_mul_fmpz_vec(fmpz * c, const fmpz_mat_t A, const fmpz * b, slong blen)
              void fmpz_mat_mul_fmpz_vec_ptr(fmpz * const * c, const fmpz_mat_t A, const fmpz * const * b, slong blen)

    Compute a matrix-vector product of ``A`` and ``(b, blen)`` and store the result in ``c``.
    The vector ``(b, blen)`` is either truncated or zero-extended to the number of columns of ``A``.
    The number of entries written to ``c`` is always equal to the number of rows of ``A``.

.. function:: void fmpz_mat_fmpz_vec_mul(fmpz * c, const fmpz * a, slong alen, const fmpz_mat_t B)
              void fmpz_mat_fmpz_vec_mul_ptr(fmpz * const * c, const fmpz * const * a, slong alen, const fmpz_mat_t B)

    Compute a vector-matrix product of ``(a, alen)`` and ``B`` and store the result in ``c``.
    The vector ``(a, alen)`` is either truncated or zero-extended to the number of rows of ``B``.
    The number of entries written to ``c`` is always equal to the number of columns of ``B``.


Inverse
--------------------------------------------------------------------------------


.. function:: int fmpz_mat_inv(fmpz_mat_t Ainv, fmpz_t den, const fmpz_mat_t A)

    Sets (``Ainv``, ``den``) to the inverse matrix of ``A``.
    Returns 1 if ``A`` is nonsingular and 0 if ``A`` is singular.
    Aliasing of ``Ainv`` and ``A`` is allowed.

    The denominator is not guaranteed to be minimal, but is guaranteed
    to be a divisor of the determinant of ``A``.

    This function uses a direct formula for matrices of size two or less,
    and otherwise solves for the identity matrix using
    fraction-free LU decomposition.



Kronecker product
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_kronecker_product(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)

    Sets ``C`` to the Kronecker product of ``A`` and ``B``.



Content
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_content(fmpz_t mat_gcd, const fmpz_mat_t A)

    Sets ``mat_gcd`` as the gcd of all the elements of the matrix ``A``.
    Returns 0 if the matrix is empty.



Trace
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_trace(fmpz_t trace, const fmpz_mat_t mat)

    Computes the trace of the matrix, i.e. the sum of the entries on
    the main diagonal. The matrix is required to be square.



Determinant
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_det(fmpz_t det, const fmpz_mat_t A)

    Sets ``det`` to the determinant of the square matrix `A`.
    The matrix of dimension `0 \times 0` is defined to have determinant 1.

    This function automatically chooses between :func:`fmpz_mat_det_cofactor`,
    :func:`fmpz_mat_det_bareiss`, :func:`fmpz_mat_det_modular` and
    :func:`fmpz_mat_det_modular_accelerated`
    (with ``proved`` = 1), depending on the size of the matrix
    and its entries.

.. function:: void fmpz_mat_det_cofactor(fmpz_t det, const fmpz_mat_t A)

    Sets ``det`` to the determinant of the square matrix `A`
    computed using direct cofactor expansion. This function only
    supports matrices up to size `4 \times 4`.

.. function:: void fmpz_mat_det_bareiss(fmpz_t det, const fmpz_mat_t A)

    Sets ``det`` to the determinant of the square matrix `A`
    computed using the Bareiss algorithm. A copy of the input matrix is
    row reduced using fraction-free Gaussian elimination, and the
    determinant is read off from the last element on the main
    diagonal.

.. function:: void fmpz_mat_det_modular(fmpz_t det, const fmpz_mat_t A, int proved)

    Sets ``det`` to the determinant of the square matrix `A`
    (if ``proved`` = 1), or a probabilistic value for the
    determinant (``proved`` = 0), computed using a multimodular
    algorithm.

    The determinant is computed modulo several small primes and
    reconstructed using the Chinese Remainder Theorem.
    With ``proved`` = 1, sufficiently many primes are chosen
    to satisfy the bound computed by ``fmpz_mat_det_bound``.
    With ``proved`` = 0, the determinant is considered determined
    if it remains unchanged modulo several consecutive primes
    (currently if their product exceeds `2^{100}`).

.. function:: void fmpz_mat_det_modular_accelerated(fmpz_t det, const fmpz_mat_t A, int proved)

    Sets ``det`` to the determinant of the square matrix `A`
    (if ``proved`` = 1), or a probabilistic value for the
    determinant (``proved`` = 0), computed using a multimodular
    algorithm.

    This function uses the same basic algorithm as ``fmpz_mat_det_modular``,
    but instead of computing `\det(A)` directly, it generates a divisor `d`
    of `\det(A)` and then computes `x = \det(A) / d` modulo several
    small primes not dividing `d`. This typically accelerates the
    computation by requiring fewer primes for large matrices, since `d`
    with high probability will be nearly as large as the determinant.
    This trick is described in [AbbottBronsteinMulders1999]_.

.. function:: void fmpz_mat_det_modular_given_divisor(fmpz_t det, const fmpz_mat_t A, const fmpz_t d, int proved)

    Given a positive divisor `d` of `\det(A)`, sets ``det`` to the
    determinant of the square matrix `A` (if ``proved`` = 1), or a
    probabilistic value for the determinant (``proved`` = 0), computed
    using a multimodular algorithm.

.. function:: void fmpz_mat_det_bound(fmpz_t bound, const fmpz_mat_t A)

    Sets ``bound`` to a nonnegative integer `B` such that
    `|\det(A)| \le B`. Assumes `A` to be a square matrix.
    The bound is computed from the Hadamard inequality
    `|\det(A)| \le \prod \|a_i\|_2` where the product is taken
    over the rows `a_i` of `A`.

.. function:: void fmpz_mat_det_bound_nonzero(fmpz_t bound, const fmpz_mat_t A)

    As per ``fmpz_mat_det_bound()`` but excludes zero columns. For use with
    non-square matrices.

.. function:: void fmpz_mat_det_divisor(fmpz_t d, const fmpz_mat_t A)

    Sets `d` to some positive divisor of the determinant of the given
    square matrix `A`, if the determinant is nonzero. If `|\det(A)| = 0`,
    `d` will always be set to zero.

    A divisor is obtained by solving `Ax = b` for an arbitrarily chosen
    right-hand side `b` using Dixon's algorithm and computing the least
    common multiple of the denominators in `x`. This yields a divisor `d`
    such that `|\det(A)| / d` is tiny with very high probability.


Transforms
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_similarity(fmpz_mat_t A, slong r, fmpz_t d)

    Applies a similarity transform to the `n\times n` matrix `M` in-place.

    If `P` is the `n\times n` identity matrix the zero entries of whose row
    `r` (`0`-indexed) have been replaced by `d`, this transform is equivalent
    to `M = P^{-1}MP`.

    Similarity transforms preserve the determinant, characteristic polynomial
    and minimal polynomial.


Characteristic polynomial
--------------------------------------------------------------------------------


.. function:: void _fmpz_mat_charpoly_berkowitz(fmpz * cp, const fmpz_mat_t mat)

    Sets ``(cp, n+1)`` to the characteristic polynomial of
    an `n \times n` square matrix.

.. function:: void fmpz_mat_charpoly_berkowitz(fmpz_poly_t cp, const fmpz_mat_t mat)

    Computes the characteristic polynomial of length `n + 1` of
    an `n \times n` square matrix. Uses an `O(n^4)` algorithm based on the
    method of Berkowitz.

.. function:: void _fmpz_mat_charpoly_modular(fmpz * cp, const fmpz_mat_t mat)

    Sets ``(cp, n+1)`` to the characteristic polynomial of
    an `n \times n` square matrix.

.. function:: void fmpz_mat_charpoly_modular(fmpz_poly_t cp, const fmpz_mat_t mat)

    Computes the characteristic polynomial of length `n + 1` of
    an `n \times n` square matrix. Uses a modular method based on an `O(n^3)`
    method over `\mathbb{Z}/n\mathbb{Z}`.

.. function:: void _fmpz_mat_charpoly(fmpz * cp, const fmpz_mat_t mat)

    Sets ``(cp, n+1)`` to the characteristic polynomial of
    an `n \times n` square matrix.

.. function:: void fmpz_mat_charpoly(fmpz_poly_t cp, const fmpz_mat_t mat)

    Computes the characteristic polynomial of length `n + 1` of
    an `n \times n` square matrix.


Minimal polynomial
--------------------------------------------------------------------------------


.. function:: slong _fmpz_mat_minpoly_modular(fmpz * cp, const fmpz_mat_t mat)

    Sets ``(cp, n+1)`` to the modular polynomial of
    an `n \times n` square matrix and returns its length.

.. function:: void fmpz_mat_minpoly_modular(fmpz_poly_t cp, const fmpz_mat_t mat)

    Computes the minimal polynomial of an `n \times n` square matrix.
    Uses a modular method based on an average time `O(n^3)`, worst case
    `O(n^4)` method over `\mathbb{Z}/n\mathbb{Z}`.

.. function:: slong _fmpz_mat_minpoly(fmpz * cp, const fmpz_mat_t mat)

    Sets ``cp`` to the minimal polynomial of an `n \times n` square
    matrix and returns its length.

.. function:: void fmpz_mat_minpoly(fmpz_poly_t cp, const fmpz_mat_t mat)

    Computes the minimal polynomial of an `n \times n` square matrix.


Rank
--------------------------------------------------------------------------------


.. function:: slong fmpz_mat_rank(const fmpz_mat_t A)

    Returns the rank, that is, the number of linearly independent columns
    (equivalently, rows), of `A`. The rank is computed by row reducing
    a copy of `A`.


Column partitioning
--------------------------------------------------------------------------------


.. function:: int fmpz_mat_col_partition(slong * part, fmpz_mat_t M, int short_circuit)

    Returns the number `p` of distinct columns of `M` (or `0` if the flag
    ``short_circuit`` is set and this number is greater than the number
    of rows of `M`). The entries of array ``part`` are set to values in
    `[0, p)` such that two entries of part are equal iff the corresponding
    columns of `M` are equal. This function is used in van Hoeij polynomial
    factoring.


Nonsingular solving
--------------------------------------------------------------------------------

The following functions allow solving matrix-matrix equations `AX = B`
where the system matrix `A` is square and has full rank. The solving
is implicitly done over the field of rational numbers: except
where otherwise noted, an integer matrix `\hat X` and a separate
denominator `d` (``den``) are computed such that `A(\hat X/d) = b`,
equivalently such that `A\hat X = bd` holds over the integers.
No guarantee is made that the numerators and denominator
are reduced to lowest terms, but the denominator is always guaranteed
to be a divisor of the determinant of `A`. If `A` is singular,
``den`` will be set to zero and the elements of the solution
vector or matrix will have undefined values. No aliasing is
allowed between arguments.

.. function:: int fmpz_mat_solve(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    This function uses Cramer's rule for small systems and
    fraction-free LU decomposition followed by fraction-free forward
    and back substitution for larger systems.

    Note that for very large systems, it is faster to compute a modular
    solution using ``fmpz_mat_solve_dixon``.

.. function:: int fmpz_mat_solve_fflu(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    Uses fraction-free LU decomposition followed by fraction-free
    forward and back substitution.

.. function:: int fmpz_mat_solve_fflu_precomp(fmpz_mat_t X, const slong * perm, const fmpz_mat_t FFLU, const fmpz_mat_t B)

    Performs fraction-free forward and back substitution given a precomputed
    fraction-free LU decomposition and corresponding permutation. If no
    impossible division is encountered, the function returns `1`. This does not
    mean the system has a solution, however a return value of `0` can only
    occur if the system is insoluble.

    If the return value is `1` and `r` is the rank of the matrix `A` whose FFLU
    we have, then the first `r` rows of `p(A)y = p(b)d` hold, where `d` is the
    denominator of the FFLU. The remaining rows must be checked by the caller.

.. function:: int fmpz_mat_solve_cramer(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.

    Uses Cramer's rule. Only systems of size up to `3 \times 3` are allowed.

.. function:: void fmpz_mat_solve_bound(fmpz_t N, fmpz_t D, const fmpz_mat_t A, const fmpz_mat_t B)

    Assuming that `A` is nonsingular, computes integers `N` and `D`
    such that the reduced numerators and denominators `n/d` in
    `A^{-1} B` satisfy the bounds `0 \le |n| \le N` and `0 \le d \le D`.

.. function:: int fmpz_mat_solve_dixon(fmpz_mat_t X, fmpz_t M, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves `AX = B` given a nonsingular square matrix `A` and a matrix `B` of
    compatible dimensions, using a modular algorithm. In particular,
    Dixon's p-adic lifting algorithm is used (currently a non-adaptive version).
    This is generally the preferred method for large dimensions.

    More precisely, this function computes an integer `M` and an integer
    matrix `X` such that `AX = B \bmod M` and such that all the reduced
    numerators and denominators of the elements `x = p/q` in the full
    solution satisfy `2|p|q < M`. As such, the explicit rational solution
    matrix can be recovered uniquely by passing the output of this
    function to ``fmpq_mat_set_fmpz_mat_mod``.

    A nonzero value is returned if `A` is nonsingular. If `A` is singular,
    zero is returned and the values of the output variables will be
    undefined.

    Aliasing between input and output matrices is allowed.


.. function:: void _fmpz_mat_solve_dixon_den(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B, const nmod_mat_t Ainv, mp_limb_t p, const fmpz_t N, const fmpz_t D)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}` using a
    ``p``-adic algorithm for the supplied prime ``p``. The values ``N`` and
    ``D`` are absolute value bounds for the numerator and denominator of the
    solution.

    Uses the Dixon lifting algorithm with early termination once the lifting
    stabilises.

.. function:: int fmpz_mat_solve_dixon_den(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    Uses the Dixon lifting algorithm with early termination once the lifting
    stabilises.

.. function:: int fmpz_mat_solve_multi_mod_den(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    Uses a Chinese remainder algorithm with early termination once the lifting
    stabilises.

.. function:: int fmpz_mat_can_solve_multi_mod_den(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Returns `1` if the system `AX = B` can be solved. If so it computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`. The
    computed denominator will not generally be minimal.

    Uses a Chinese remainder algorithm.

    Note that the matrices `A` and `B` may have any shape as long as they have
    the same number of rows.

.. function:: int fmpz_mat_can_solve_fflu(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Returns `1` if the system `AX = B` can be solved. If so it computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`. The
    computed denominator will not generally be minimal.

    Uses a fraction free LU decomposition algorithm.

    Note that the matrices `A` and `B` may have any shape as long as they have
    the same number of rows.

.. function:: int fmpz_mat_can_solve(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_mat_t B)

    Returns `1` if the system `AX = B` can be solved. If so it computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`. The
    computed denominator will not generally be minimal.

    Note that the matrices `A` and `B` may have any shape as long as they have
    the same number of rows.

Row reduction
--------------------------------------------------------------------------------


.. function:: slong fmpz_mat_find_pivot_any(const fmpz_mat_t mat, slong start_row, slong end_row, slong c)

    Attempts to find a pivot entry for row reduction.
    Returns a row index `r` between ``start_row`` (inclusive) and
    ``stop_row`` (exclusive) such that column `c` in ``mat`` has
    a nonzero entry on row `r`, or returns -1 if no such entry exists.

    This implementation simply chooses the first nonzero entry
    it encounters. This is likely to be a nearly optimal choice if all
    entries in the matrix have roughly the same size, but can lead to
    unnecessary coefficient growth if the entries vary in size.

.. function:: slong fmpz_mat_fflu(fmpz_mat_t B, fmpz_t den, slong * perm, const fmpz_mat_t A, int rank_check)

    Uses fraction-free Gaussian elimination to set (``B``, ``den``) to a
    fraction-free LU decomposition of ``A`` and returns the
    rank of ``A``. Aliasing of ``A`` and ``B`` is allowed.

    Pivot elements are chosen with ``fmpz_mat_find_pivot_any``.
    If ``perm`` is non-``NULL``, the permutation of
    rows in the matrix will also be applied to ``perm``.

    If ``rank_check`` is set, the function aborts and returns 0 if the
    matrix is detected not to have full rank without completing the
    elimination.

    The denominator ``den`` is set to `\pm \operatorname{det}(S)` where
    `S` is an appropriate submatrix of `A` (`S = A` if `A` is square)
    and the sign is decided by the parity of the permutation. Note that the
    determinant is not generally the minimal denominator.

    The fraction-free LU decomposition is defined in [NakTurWil1997]_.

.. function:: slong fmpz_mat_rref(fmpz_mat_t B, fmpz_t den, const fmpz_mat_t A)

    Sets (``B``, ``den``) to the reduced row echelon form of ``A``
    and returns the rank of ``A``. Aliasing of ``A`` and ``B``
    is allowed.

    The algorithm used chooses between ``fmpz_mat_rref_fflu`` and
    ``fmpz_mat_rref_mul`` based on the dimensions of the input matrix.

.. function:: slong fmpz_mat_rref_fflu(fmpz_mat_t B, fmpz_t den, const fmpz_mat_t A)

    Sets (``B``, ``den``) to the reduced row echelon form of ``A``
    and returns the rank of ``A``. Aliasing of ``A`` and ``B``
    is allowed.

    The algorithm proceeds by first computing a row echelon form using
    ``fmpz_mat_fflu``. Letting the upper part of this matrix be
    `(U | V) P` where `U` is full rank upper triangular and `P` is a
    permutation matrix, we obtain the rref by setting `V` to `U^{-1} V`
    using back substitution. Scaling each completed row in the back
    substitution to the denominator ``den``, we avoid introducing
    new fractions. This strategy is equivalent to the fraction-free
    Gauss-Jordan elimination in [NakTurWil1997]_, but faster since
    only the part `V` corresponding to the null space has to be updated.

    The denominator ``den`` is set to `\pm \operatorname{det}(S)` where
    `S` is an appropriate submatrix of `A` (`S = A` if `A` is square).
    Note that the determinant is not generally the minimal denominator.

.. function:: slong fmpz_mat_rref_mul(fmpz_mat_t B, fmpz_t den, const fmpz_mat_t A)

    Sets (``B``, ``den``) to the reduced row echelon form of ``A``
    and returns the rank of ``A``. Aliasing of ``A`` and ``B``
    is allowed.

    The algorithm works by computing the reduced row echelon form of ``A``
    modulo a prime `p` using ``nmod_mat_rref``. The pivot columns and rows
    of this matrix will then define a non-singular submatrix of ``A``,
    nonsingular solving and matrix multiplication can then be used to determine
    the reduced row echelon form of the whole of ``A``. This procedure is
    described in [Stein2007]_.

.. function:: int fmpz_mat_is_in_rref_with_rank(const fmpz_mat_t A, const fmpz_t den, slong rank)

    Checks that the matrix `A/den` is in reduced row echelon form of rank
    ``rank``, returns 1 if so and 0 otherwise.


Modular gaussian elimination
--------------------------------------------------------------------------------


.. function:: slong fmpz_mat_rref_mod(slong * perm, fmpz_mat_t A, const fmpz_t p)

    Uses fraction-free Gauss-Jordan elimination to set ``A``
    to its reduced row echelon form and returns the rank of ``A``.
    All computations are done modulo p.

    Pivot elements are chosen with ``fmpz_mat_find_pivot_any``.
    If ``perm`` is non-``NULL``, the permutation of
    rows in the matrix will also be applied to ``perm``.


Strong echelon form and Howell form
--------------------------------------------------------------------------------


.. function:: void fmpz_mat_strong_echelon_form_mod(fmpz_mat_t A, const fmpz_t mod)

    Transforms `A` such that `A` modulo ``mod`` is the strong echelon form
    of the input matrix modulo ``mod``. The Howell form and the strong
    echelon form are equal up to permutation of the rows, see [FieHof2014]_
    for a definition of the strong echelon form and the algorithm used here.

    `A` must have at least as many rows as columns.

.. function:: slong fmpz_mat_howell_form_mod(fmpz_mat_t A, const fmpz_t mod)

    Transforms `A` such that `A` modulo ``mod`` is the Howell form of the
    input matrix modulo ``mod``.
    For a definition of the Howell form see [StoMul1998]_. The Howell form
    is computed by first putting `A` into strong echelon form and then ordering
    the rows.

    `A` must have at least as many rows as columns.


Nullspace
--------------------------------------------------------------------------------


.. function:: slong fmpz_mat_nullspace(fmpz_mat_t B, const fmpz_mat_t A)

    Computes a basis for the right rational nullspace of `A` and returns
    the dimension of the nullspace (or nullity). `B` is set to a matrix with
    linearly independent columns and maximal rank such that `AB = 0`
    (i.e. `Ab = 0` for each column `b` in `B`), and the rank of `B` is
    returned.

    In general, the entries in `B` will not be minimal: in particular,
    the pivot entries in `B` will generally differ from unity.
    `B` must be allocated with sufficient space to represent the result
    (at most `n \times n` where `n` is the number of columns of `A`).



Echelon form
--------------------------------------------------------------------------------


.. function:: slong fmpz_mat_rref_fraction_free(slong * perm, fmpz_mat_t B, fmpz_t den, const fmpz_mat_t A)

    Computes an integer matrix ``B`` and an integer ``den`` such that
    ``B / den`` is the unique row reduced echelon form (RREF) of ``A``
    and returns the rank, i.e. the number of nonzero rows in ``B``.

    Aliasing of ``B`` and ``A`` is allowed, with an in-place
    computation being more efficient. The size of ``B`` must be
    the same as that of ``A``.

    The permutation order will be written to ``perm`` unless this
    argument is ``NULL``. That is, row ``i`` of the output matrix will
    correspond to row ``perm[i]`` of the input matrix.

    The denominator will always be a divisor of the determinant of (some
    submatrix of) `A`, but is not guaranteed to be minimal or canonical in
    any other sense.


Hermite normal form
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.. function:: void fmpz_mat_hnf(fmpz_mat_t H, const fmpz_mat_t A)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of ``A``. The algorithm used is selected from the
    implementations in FLINT to be the one most likely to be optimal, based on
    the characteristics of the input matrix.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_hnf_transform(fmpz_mat_t H, fmpz_mat_t U, const fmpz_mat_t A)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of ``A`` along with the transformation matrix
    ``U`` such that `UA = H`. The algorithm used is selected from the
    implementations in FLINT as per ``fmpz_mat_hnf``.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A`` and ``U`` must be square of \compatible
    dimension (having the same number of rows as ``A``).

.. function:: void fmpz_mat_hnf_classical(fmpz_mat_t H, const fmpz_mat_t A)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of ``A``. The algorithm used is straightforward and
    is described, for example, in [Algorithm 2.4.4] [Coh1996]_.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_hnf_xgcd(fmpz_mat_t H, const fmpz_mat_t A)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of ``A``. The algorithm used is an improvement on the
    basic algorithm and uses extended gcds to speed up computation, this method
    is described, for example, in [Algorithm 2.4.5] [Coh1996]_.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_hnf_modular(fmpz_mat_t H, const fmpz_mat_t A, const fmpz_t D)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of the `m\times n` matrix ``A``, where ``A`` is
    assumed to be of rank `n` and ``D`` is known to be a positive multiple of
    the determinant of the non-zero rows of ``H``. The algorithm used here is
    due to Domich, Kannan and Trotter [DomKanTro1987]_ and is also described
    in [Algorithm 2.4.8] [Coh1996]_.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_hnf_modular_eldiv(fmpz_mat_t A, const fmpz_t D)

    Transforms the `m\times n` matrix ``A`` into Hermite normal form,
    where ``A`` is assumed to be of rank `n` and ``D`` is known to be a
    positive multiple of the largest elementary divisor of ``A``.
    The algorithm used here is described in [FieHof2014]_.

.. function:: void fmpz_mat_hnf_minors(fmpz_mat_t H, const fmpz_mat_t A)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of the `m\times n` matrix ``A``, where ``A`` is
    assumed to be of rank `n`. The algorithm used here is due to Kannan and
    Bachem [KanBac1979]_ and takes the principal minors to Hermite normal
    form in turn.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_hnf_pernet_stein(fmpz_mat_t H, const fmpz_mat_t A, flint_rand_t state)

    Computes an integer matrix ``H`` such that ``H`` is the unique (row)
    Hermite normal form of the `m\times n` matrix ``A``. The algorithm used
    here is due to Pernet and Stein [PernetStein2010]_.

    Aliasing of ``H`` and ``A`` is allowed. The size of ``H`` must be
    the same as that of ``A``.

.. function:: int fmpz_mat_is_in_hnf(const fmpz_mat_t A)

    Checks that the given matrix is in Hermite normal form, returns 1 if so and
    0 otherwise.


Smith normal form
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.. function:: void fmpz_mat_snf(fmpz_mat_t S, const fmpz_mat_t A)

    Computes an integer matrix ``S`` such that ``S`` is the unique Smith
    normal form of ``A``. The algorithm used is selected from the
    implementations in FLINT to be the one most likely to be optimal, based on
    the characteristics of the input matrix.

    Aliasing of ``S`` and ``A`` is allowed. The size of ``S`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_snf_diagonal(fmpz_mat_t S, const fmpz_mat_t A)

    Computes an integer matrix ``S`` such that ``S`` is the unique Smith
    normal form of the diagonal matrix ``A``. The algorithm used simply takes
    gcds of pairs on the diagonal in turn until the Smith form is obtained.

    Aliasing of ``S`` and ``A`` is allowed. The size of ``S`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_snf_kannan_bachem(fmpz_mat_t S, const fmpz_mat_t A)

    Computes an integer matrix ``S`` such that ``S`` is the unique Smith
    normal form of the diagonal matrix ``A``. The algorithm used here is due
    to Kannan and Bachem [KanBac1979]_

    Aliasing of ``S`` and ``A`` is allowed. The size of ``S`` must be
    the same as that of ``A``.

.. function:: void fmpz_mat_snf_iliopoulos(fmpz_mat_t S, const fmpz_mat_t A, const fmpz_t mod)

    Computes an integer matrix ``S`` such that ``S`` is the unique Smith
    normal form of the nonsingular `n\times n` matrix ``A``. The algorithm
    used is due to Iliopoulos [Iliopoulos1989]_.

    Aliasing of ``S`` and ``A`` is allowed. The size of ``S`` must be
    the same as that of ``A``.

.. function:: int fmpz_mat_is_in_snf(const fmpz_mat_t A)

    Checks that the given matrix is in Smith normal form, returns 1 if so and 0
    otherwise.


Special matrices
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.. function:: void fmpz_mat_gram(fmpz_mat_t B, const fmpz_mat_t A)

    Sets ``B`` to the Gram matrix of the `m`-dimensional lattice ``L`` in
    `n`-dimensional Euclidean space `R^n` spanned by the rows of
    the `m \times n` matrix ``A``. Dimensions must be compatible.
    ``A`` and ``B`` are allowed to be the same object if ``A`` is a
    square matrix.

.. function:: int fmpz_mat_is_hadamard(const fmpz_mat_t H)

    Returns nonzero iff `H` is a Hadamard matrix, meaning
    that it is a square matrix, only has entries that are `\pm 1`,
    and satisfies `H^T = n H^{-1}` where `n` is the matrix size.

.. function:: int fmpz_mat_hadamard(fmpz_mat_t H)

    Attempts to set the matrix `H` to a Hadamard matrix, returning 1 if
    successful and 0 if unsuccessful.

    A Hadamard matrix of size `n` can only exist if `n` is 1, 2,
    or a multiple of 4. It is not known whether a
    Hadamard matrix exists for every size that is a multiple of 4.
    This function uses the Paley construction, which
    succeeds for all `n` of the form `n = 2^e` or `n = 2^e (q + 1)` where
    `q` is an odd prime power. Orders `n` for which Hadamard matrices are
    known to exist but for which this construction fails are
    92, 116, 156, ... (OEIS A046116).


Conversions
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.. function:: int fmpz_mat_get_d_mat(d_mat_t B, const fmpz_mat_t A)

    Sets the entries of ``B`` as doubles corresponding to the entries of
    ``A``, rounding down towards zero if the latter cannot be represented
    exactly. The return value is -1 if any entry of ``A`` is too large to
    fit in the normal range of a double, and 0 otherwise.

.. function:: int fmpz_mat_get_d_mat_transpose(d_mat_t B, const fmpz_mat_t A)

    Sets the entries of ``B`` as doubles corresponding to the entries of
    the transpose of ``A``, rounding down towards zero if the latter cannot
    be represented exactly. The return value is -1 if any entry of ``A`` is
    too large to fit in the normal range of a double, and 0 otherwise.


Cholesky Decomposition
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.. function:: void fmpz_mat_chol_d(d_mat_t R, const fmpz_mat_t A)

    Computes ``R``, the Cholesky factor of a symmetric, positive definite
    matrix ``A`` using the Cholesky decomposition process. (Sets ``R``
    such that `A = RR^{T}` where ``R`` is a lower triangular matrix.)


LLL
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.. function:: int fmpz_mat_is_reduced(const fmpz_mat_t A, double delta, double eta)
              int fmpz_mat_is_reduced_gram(const fmpz_mat_t A, double delta, double eta)

    Returns a non-zero value if the basis ``A`` is LLL-reduced with factor
    (``delta``, ``eta``), and otherwise returns zero.
    The second version assumes ``A`` is the Gram matrix of the basis.

.. function:: int fmpz_mat_is_reduced_with_removal(const fmpz_mat_t A, double delta, double eta, const fmpz_t gs_B, int newd)
              int fmpz_mat_is_reduced_gram_with_removal(const fmpz_mat_t A, double delta, double eta, const fmpz_t gs_B, int newd)

    Returns a non-zero value if the basis ``A`` is LLL-reduced with factor
    (``delta``, ``eta``) for each of the first ``newd`` vectors and the squared
    Gram-Schmidt length of each of the remaining `i`-th vectors
    (where `i \ge` ``newd``) is greater than ``gs_B``, and otherwise returns zero.
    The second version assumes ``A`` is the Gram matrix of the basis.


Classical LLL
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.. function:: void fmpz_mat_lll_original(fmpz_mat_t A, const fmpq_t delta, const fmpq_t eta)

    Takes a basis `x_1, x_2, \ldots, x_m` of the lattice `L \subset R^n` (as
    the rows of a `m \times n` matrix ``A``). The output is a (``delta``,
    ``eta``)-reduced basis `y_1, y_2, \ldots, y_m` of the lattice `L` (as
    the rows of the same `m \times n` matrix ``A``).


Modified LLL
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.. function:: void fmpz_mat_lll_storjohann(fmpz_mat_t A, const fmpq_t delta, const fmpq_t eta)

    Takes a basis `x_1, x_2, \ldots, x_m` of the lattice `L \subset R^n` (as
    the rows of a `m \times n` matrix ``A``). The output is an (``delta``,
    ``eta``)-reduced basis `y_1, y_2, \ldots, y_m` of the lattice `L` (as
    the rows of the same `m \times n` matrix ``A``). Uses a modified version of
    LLL, which has better complexity in terms of the lattice dimension,
    introduced by Storjohann.

    See "Faster Algorithms for Integer Lattice Basis Reduction." Technical
    Report 249. Zurich, Switzerland: Department Informatik, ETH. July 30,
    1996.
