.. _fmpq-mat:

**fmpq_mat.h** -- matrices over the rational numbers
===============================================================================

The :type:`fmpq_mat_t` data type represents matrices over `\mathbb{Q}`.

A rational matrix is stored as an array of ``fmpq`` elements in order
to allow convenient and efficient manipulation of individual entries.
In general, ``fmpq_mat`` functions assume that input entries are in
canonical form, and produce output with entries in canonical form.

Since rational arithmetic is expensive, computations are typically
performed by clearing denominators, performing the heavy work over the
integers, and converting the final result back to a rational matrix.
The ``fmpq_mat`` functions take care of such conversions
transparently. For users who need fine-grained control, various
functions for conversion between rational and integer matrices are
provided.

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

.. type:: fmpq_mat_struct

.. type:: fmpq_mat_t

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


.. function:: void fmpq_mat_init(fmpq_mat_t mat, slong rows, slong cols)

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

.. function:: void fmpq_mat_init_set(fmpq_mat_t mat1, const fmpq_mat_t mat2)

    Initialises ``mat1`` and sets it equal to ``mat2``.

.. function:: void fmpq_mat_clear(fmpq_mat_t mat)

    Frees all memory associated with the matrix. The matrix must be
    reinitialised if it is to be used again.

.. function:: void fmpq_mat_swap(fmpq_mat_t mat1, fmpq_mat_t mat2)

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

.. function:: void fmpq_mat_swap_entrywise(fmpq_mat_t mat1, fmpq_mat_t mat2)

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


Entry access
--------------------------------------------------------------------------------


.. function:: fmpq * fmpq_mat_entry(const fmpq_mat_t mat, slong i, slong j)

    Gives a reference to the entry at row ``i`` and column ``j``.
    The reference can be passed as an input or output variable to any
    ``fmpq`` function for direct manipulation of the matrix element.
    No bounds checking is performed.

.. function:: fmpz * fmpq_mat_entry_num(const fmpq_mat_t mat, slong i, slong j)

    Gives a reference to the numerator of the entry at row ``i`` and
    column ``j``. The reference can be passed as an input or output
    variable to any ``fmpz`` function for direct manipulation of the
    matrix element. No bounds checking is performed.

.. function:: fmpz * fmpq_mat_entry_den(const fmpq_mat_t mat, slong i, slong j)

    Gives a reference to the denominator of the entry at row ``i`` and
    column ``j``. The reference can be passed as an input or output
    variable to any ``fmpz`` function for direct manipulation of the
    matrix element. No bounds checking is performed.

.. function:: slong fmpq_mat_nrows(const fmpq_mat_t mat)

    Return the number of rows of the matrix ``mat``.

.. function:: slong fmpq_mat_ncols(const fmpq_mat_t mat)

    Return the number of columns of the matrix ``mat``.


Basic assignment
--------------------------------------------------------------------------------


.. function:: void fmpq_mat_set(fmpq_mat_t dest, const fmpq_mat_t src)

    Sets the entries in ``dest`` to the same values as in ``src``,
    assuming the two matrices have the same dimensions.

.. function:: void fmpq_mat_zero(fmpq_mat_t mat)

    Sets ``mat`` to the zero matrix.

.. function:: void fmpq_mat_one(fmpq_mat_t mat)

    Let `m` be the minimum of the number of rows and columns
    in the matrix ``mat``.  This function sets the first
    `m \times m` block to the identity matrix, and the remaining
    block to zero.

.. function:: void fmpq_mat_transpose(fmpq_mat_t rop, const fmpq_mat_t op)

    Sets the matrix ``rop`` to the transpose of the matrix ``op``,
    assuming that their dimensions are compatible. Aliasing is allowed for square matrices.

.. function:: void fmpq_mat_swap_rows(fmpq_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 fmpq_mat_swap_cols(fmpq_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 fmpq_mat_invert_rows(fmpq_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 fmpq_mat_invert_cols(fmpq_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``.

Addition, scalar multiplication
--------------------------------------------------------------------------------


.. function:: void fmpq_mat_add(fmpq_mat_t mat, const fmpq_mat_t mat1, const fmpq_mat_t mat2)

    Sets ``mat`` to the sum of ``mat1`` and ``mat2``,
    assuming that all three matrices have the same dimensions.

.. function:: void fmpq_mat_sub(fmpq_mat_t mat, const fmpq_mat_t mat1, const fmpq_mat_t mat2)

    Sets ``mat`` to the difference of ``mat1`` and ``mat2``,
    assuming that all three matrices have the same dimensions.

.. function:: void fmpq_mat_neg(fmpq_mat_t rop, const fmpq_mat_t op)

    Sets ``rop`` to the negative of ``op``, assuming that
    the two matrices have the same dimensions.

.. function:: void fmpq_mat_scalar_mul_fmpq(fmpq_mat_t rop, const fmpq_mat_t op, const fmpq_t x)

    Sets ``rop`` to ``op`` multiplied by the rational `x`,
    assuming that the two matrices have the same dimensions.

    Note that the rational ``x`` may not be aliased with any part of the
    entries of ``rop``.

.. function:: void fmpq_mat_scalar_mul_fmpz(fmpq_mat_t rop, const fmpq_mat_t op, const fmpz_t x)

    Sets ``rop`` to ``op`` multiplied by the integer `x`,
    assuming that the two matrices have the same dimensions.

    Note that the integer `x` may not be aliased with any part of
    the entries of ``rop``.

.. function:: void fmpq_mat_scalar_div_fmpz(fmpq_mat_t rop, const fmpq_mat_t op, const fmpz_t x)

    Sets ``rop`` to ``op`` divided by the integer `x`,
    assuming that the two matrices have the same dimensions
    and that `x` is non-zero.

    Note that the integer `x` may not be aliased with any part of
    the entries of ``rop``.


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


.. function:: void fmpq_mat_print(const fmpq_mat_t mat)

    Prints the matrix ``mat`` to standard output.


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


.. function:: void fmpq_mat_randbits(fmpq_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)

    This is equivalent to applying ``fmpq_randbits`` to all entries
    in the matrix.

.. function:: void fmpq_mat_randtest(fmpq_mat_t mat, flint_rand_t state, flint_bitcnt_t bits)

    This is equivalent to applying ``fmpq_randtest`` to all entries
    in the matrix.


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


.. function:: void fmpq_mat_window_init(fmpq_mat_t window, const fmpq_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 fmpq_mat_window_clear(fmpq_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.


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


.. function:: void fmpq_mat_concat_vertical(fmpq_mat_t res, const fmpq_mat_t mat1, const fmpq_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 fmpq_mat_concat_horizontal(fmpq_mat_t res, const fmpq_mat_t mat1, const fmpq_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)`.


Special matrices
--------------------------------------------------------------------------------


.. function:: void fmpq_mat_hilbert_matrix(fmpq_mat_t mat)

    Sets ``mat`` to a Hilbert matrix of the given size. That is,
    the entry at row `i` and column `j` is set to `1/(i+j+1)`.


Basic comparison and properties
--------------------------------------------------------------------------------


.. function:: int fmpq_mat_equal(const fmpq_mat_t mat1, const fmpq_mat_t mat2)

    Returns nonzero if ``mat1`` and ``mat2`` have the same shape and
    all their entries agree, and returns zero otherwise. Assumes the
    entries in both ``mat1`` and ``mat2`` are in canonical form.

.. function:: int fmpq_mat_is_integral(const fmpq_mat_t mat)

    Returns nonzero if all entries in ``mat`` are integer-valued, and
    returns zero otherwise. Assumes that the entries in ``mat``
    are in canonical form.

.. function:: int fmpq_mat_is_zero(const fmpq_mat_t mat)

    Returns nonzero if all entries in ``mat`` are zero, and returns
    zero otherwise.

.. function:: int fmpq_mat_is_one(const fmpq_mat_t mat)

    Returns nonzero if ``mat`` ones along the diagonal and zeros elsewhere,
    and returns zero otherwise.

.. function:: int fmpq_mat_is_empty(const fmpq_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 fmpq_mat_is_square(const fmpq_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.



Integer matrix conversion
--------------------------------------------------------------------------------


.. function:: int fmpq_mat_get_fmpz_mat(fmpz_mat_t dest, const fmpq_mat_t mat)

    Sets ``dest`` to ``mat`` and returns nonzero if all entries
    in ``mat`` are integer-valued. If not all entries in ``mat``
    are integer-valued, sets ``dest`` to an undefined matrix
    and returns zero. Assumes that the entries in ``mat`` are
    in canonical form.

.. function:: void fmpq_mat_get_fmpz_mat_entrywise(fmpz_mat_t num, fmpz_mat_t den, const fmpq_mat_t mat)

    Sets the integer matrices ``num`` and ``den`` respectively
    to the numerators and denominators of the entries in ``mat``.

.. function:: void fmpq_mat_get_fmpz_mat_matwise(fmpz_mat_t num, fmpz_t den, const fmpq_mat_t mat)

    Converts all entries in ``mat`` to a common denominator,
    storing the rescaled numerators in ``num`` and the
    denominator in ``den``. The denominator will be minimal
    if the entries in ``mat`` are in canonical form.

.. function:: void fmpq_mat_get_fmpz_mat_rowwise(fmpz_mat_t num, fmpz * den, const fmpq_mat_t mat)

    Clears denominators in ``mat`` row by row. The rescaled
    numerators are written to ``num``, and the denominator
    of row ``i`` is written to position ``i`` in ``den``
    which can be a preinitialised ``fmpz`` vector. Alternatively,
    ``NULL`` can be passed as the ``den`` variable, in which
    case the denominators will not be stored.

.. function:: void fmpq_mat_get_fmpz_mat_rowwise_2(fmpz_mat_t num, fmpz_mat_t num2, fmpz * den, const fmpq_mat_t mat, const fmpq_mat_t mat2)

    Clears denominators row by row of both ``mat`` and ``mat2``,
    writing the respective numerators to ``num`` and ``num2``.
    This is equivalent to concatenating ``mat`` and ``mat2``
    horizontally, calling ``fmpq_mat_get_fmpz_mat_rowwise``,
    and extracting the two submatrices in the result.

.. function:: void fmpq_mat_get_fmpz_mat_colwise(fmpz_mat_t num, fmpz * den, const fmpq_mat_t mat)

    Clears denominators in ``mat`` column by column. The rescaled
    numerators are written to ``num``, and the denominator
    of column ``i`` is written to position ``i`` in ``den``
    which can be a preinitialised ``fmpz`` vector. Alternatively,
    ``NULL`` can be passed as the ``den`` variable, in which
    case the denominators will not be stored.

.. function:: void fmpq_mat_set_fmpz_mat(fmpq_mat_t dest, const fmpz_mat_t src)

    Sets ``dest`` to ``src``.

.. function:: void fmpq_mat_set_fmpz_mat_div_fmpz(fmpq_mat_t mat, const fmpz_mat_t num, const fmpz_t den)

    Sets ``mat`` to the integer matrix ``num`` divided by the
    common denominator ``den``.


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


.. function:: void fmpq_mat_get_fmpz_mat_mod_fmpz(fmpz_mat_t dest, const fmpq_mat_t mat, const fmpz_t mod)

    Sets each entry in ``dest`` to the corresponding entry in ``mat``,
    reduced modulo ``mod``.

.. function:: int fmpq_mat_set_fmpz_mat_mod_fmpz(fmpq_mat_t X, const fmpz_mat_t Xmod, const fmpz_t mod)

    Sets ``X`` to the entrywise rational reconstruction integer matrix
    ``Xmod`` modulo ``mod``, and returns nonzero if the reconstruction
    is successful. If rational reconstruction fails for any element,
    returns zero and sets the entries in ``X`` to undefined values.


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


.. function:: void fmpq_mat_mul_direct(fmpq_mat_t C, const fmpq_mat_t A, const fmpq_mat_t B)

    Sets ``C`` to the matrix product ``AB``, computed
    naively using rational arithmetic. This is typically very slow and
    should only be used in circumstances where clearing denominators
    would consume too much memory.

.. function:: void fmpq_mat_mul_cleared(fmpq_mat_t C, const fmpq_mat_t A, const fmpq_mat_t B)

    Sets ``C`` to the matrix product ``AB``, computed
    by clearing denominators and multiplying over the integers.

.. function:: void fmpq_mat_mul(fmpq_mat_t C, const fmpq_mat_t A, const fmpq_mat_t B)

    Sets ``C`` to the matrix product ``AB``. This
    simply calls ``fmpq_mat_mul_cleared``.

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

    Sets ``C`` to the matrix product ``AB``, with ``B``
    an integer matrix. This function works efficiently by clearing
    denominators of ``A``.

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

    Sets ``C`` to the matrix product ``AB``, with ``A``
    an integer matrix. This function works efficiently by clearing
    denominators of ``B``.

.. function:: void fmpq_mat_mul_fmpq_vec(fmpq * c, const fmpq_mat_t A, const fmpq * b, slong blen)
              void fmpq_mat_mul_fmpz_vec(fmpq * c, const fmpq_mat_t A, const fmpz * b, slong blen)
              void fmpq_mat_mul_fmpq_vec_ptr(fmpq * const * c, const fmpq_mat_t A, const fmpq * const * b, slong blen)
              void fmpq_mat_mul_fmpz_vec_ptr(fmpq * const * c, const fmpq_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 entries written to ``c`` is always equal to the number of rows of ``A``.

.. function:: void fmpq_mat_fmpq_vec_mul(fmpq * c, const fmpq * a, slong alen, const fmpq_mat_t B)
              void fmpq_mat_fmpz_vec_mul(fmpq * c, const fmpz * a, slong alen, const fmpq_mat_t B)
              void fmpq_mat_fmpq_vec_mul_ptr(fmpq * const * c, const fmpq * const * a, slong alen, const fmpq_mat_t B)
              void fmpq_mat_fmpz_vec_mul_ptr(fmpq * const * c, const fmpz * const * a, slong alen, const fmpq_mat_t B)

    Compute a vector-matrix product of ``(a, alen)`` and ``B`` and 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 entries written to ``c`` is always equal to the number of columns of ``B``.


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


.. function:: void fmpq_mat_kronecker_product(fmpq_mat_t C, const fmpq_mat_t A, const fmpq_mat_t B)

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


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


.. function:: void fmpq_mat_trace(fmpq_t trace, const fmpq_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 fmpq_mat_det(fmpq_t det, const fmpq_mat_t mat)

    Sets ``det`` to the determinant of ``mat``. In the general case,
    the determinant is computed by clearing denominators and computing a
    determinant over the integers. Matrices of size 0, 1 or 2 are handled
    directly.

Permanent
--------------------------------------------------------------------------------


.. function:: int fmpq_mat_permanent(fmpq_t res, const fmpq_mat_t A)

    Sets ``res`` to the permanent of the square matrix *A*, returning 1
    on success. If the matrix is too large, returns 0.


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


.. function:: int fmpq_mat_solve_fraction_free(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)
              int fmpq_mat_solve_dixon(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)
              int fmpq_mat_solve_multi_mod(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)
              int fmpq_mat_solve(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)

    Solves ``AX = B`` for nonsingular ``A``.
    Returns nonzero if ``A`` is nonsingular or if the right hand side
    is empty, and zero otherwise.

    All algorithms clear denominators to obtain a rescaled system over the integers.
    The *fraction_free* algorithm uses FFLU solving over the integers.
    The *dixon* and *multi_mod* algorithms use Dixon p-adic lifting
    or multimodular solving, followed by rational reconstruction
    with an adaptive stopping test. The *dixon* and *multi_mod* algorithms
    are generally the best choice for large systems.

    The default method chooses an algorithm automatically.

.. function:: int fmpq_mat_solve_fmpz_mat_fraction_free(fmpq_mat_t X, const fmpz_mat_t A, const fmpz_mat_t B)
              int fmpq_mat_solve_fmpz_mat_dixon(fmpq_mat_t X, const fmpz_mat_t A, const fmpz_mat_t B)
              int fmpq_mat_solve_fmpz_mat_multi_mod(fmpq_mat_t X, const fmpz_mat_t A, const fmpz_mat_t B)
              int fmpq_mat_solve_fmpz_mat(fmpq_mat_t X, const fmpz_mat_t A, const fmpz_mat_t B)

    Solves ``AX = B`` for nonsingular ``A``, where *A* and *B* are integer
    matrices. Returns nonzero if ``A`` is nonsingular or if the right hand side
    is empty, and zero otherwise.


.. function:: int fmpq_mat_can_solve_multi_mod(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)

    Returns `1` if ``AX = B`` has a solution and if so, sets ``X`` to one such
    solution. The matrices can have any shape but must have the same number of
    rows.

.. function:: int fmpq_mat_can_solve_fraction_free(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)

    Returns `1` if ``AX = B`` has a solution and if so, sets ``X`` to one such
    solution. The matrices can have any shape but must have the same number of
    rows.

int fmpq_mat_can_solve_fmpz_mat_dixon(fmpq_mat_t X, const fmpz_mat_t A, const fmpz_mat_t B)

    Returns `1` if ``AX = B`` has a solution and if so, sets ``X`` to one such
    solution. The matrices can have any shape but must have the same number of
    rows. The input matrices must have integer entries and `A` cannot be an
    empty matrix.

int fmpq_mat_can_solve_dixon(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)

    Returns `1` if ``AX = B`` has a solution and if so, sets ``X`` to one such
    solution. The matrices can have any shape but must have the same number of
    rows.

.. function:: int fmpq_mat_can_solve(fmpq_mat_t X, const fmpq_mat_t A, const fmpq_mat_t B)

    Returns `1` if ``AX = B`` has a solution and if so, sets ``X`` to one such
    solution. The matrices can have any shape but must have the same number of
    rows.


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


.. function:: int fmpq_mat_inv(fmpq_mat_t B, const fmpq_mat_t A)

    Sets ``B`` to the inverse matrix of ``A`` and returns nonzero.
    Returns zero if ``A`` is singular. ``A`` must be a square matrix.



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


.. function:: int fmpq_mat_pivot(slong * perm, fmpq_mat_t mat, slong r, slong c)

    Helper function for row reduction. Returns 1 if the entry of ``mat``
    at row `r` and column `c` is nonzero. Otherwise searches for a nonzero
    entry in the same column among rows `r+1, r+2, \ldots`. If a nonzero
    entry is found at row `s`, swaps rows `r` and `s` and the corresponding
    entries in ``perm`` (unless ``NULL``) and returns -1. If no
    nonzero pivot entry is found, leaves the inputs unchanged and returns 0.

.. function:: slong fmpq_mat_rref_classical(fmpq_mat_t B, const fmpq_mat_t A)

    Sets ``B`` to the reduced row echelon form of ``A`` and returns
    the rank. Performs Gauss-Jordan elimination directly over the rational
    numbers. This algorithm is usually inefficient and is mainly intended
    to be used for testing purposes.

.. function:: slong fmpq_mat_rref_fraction_free(fmpq_mat_t B, const fmpq_mat_t A)

    Sets ``B`` to the reduced row echelon form of ``A`` and returns
    the rank. Clears denominators and performs fraction-free Gauss-Jordan
    elimination using ``fmpz_mat`` functions.

.. function:: slong fmpq_mat_rref(fmpq_mat_t B, const fmpq_mat_t A)

    Sets ``B`` to the reduced row echelon form of ``A`` and returns
    the rank. This function automatically chooses between the classical and
    fraction-free algorithms depending on the size of the matrix.


Gram-Schmidt Orthogonalisation
--------------------------------------------------------------------------------


.. function:: void fmpq_mat_gso(fmpq_mat_t B, const fmpq_mat_t A)

    Takes a subset of `\mathbb{Q}^m` `S = \{a_1, a_2, \ldots ,a_n\}` (as the
    columns of a `m \times n` matrix ``A``) and generates an orthogonal set
    `S' = \{b_1, b_2, \ldots ,b_n\}` (as the columns of the `m \times n` matrix
    ``B``) that spans the same subspace of `\mathbb{Q}^m` as `S`.


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


.. function:: void fmpq_mat_similarity(fmpq_mat_t A, slong r, fmpq_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 _fmpq_mat_charpoly(fmpz * coeffs, fmpz_t den, const fmpq_mat_t mat)

    Set ``(coeffs, den)`` to the characteristic polynomial of the given
    `n\times n` matrix.

.. function:: void fmpq_mat_charpoly(fmpq_poly_t pol, const fmpq_mat_t mat)

    Set ``pol`` to the characteristic polynomial of the given `n\times n`
    matrix. If ``mat`` is not square, an exception is raised.


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


.. function:: slong _fmpq_mat_minpoly(fmpz * coeffs, fmpz_t den, const fmpq_mat_t mat)

    Set ``(coeffs, den)`` to the minimal polynomial of the given
    `n\times n` matrix and return the length of the polynomial.

.. function:: void fmpq_mat_minpoly(fmpq_poly_t pol, const fmpq_mat_t mat)

    Set ``pol`` to the minimal polynomial of the given `n\times n`
    matrix. If ``mat`` is not square, an exception is raised.
