# fmpq_mat.h – matrices over the rational numbers¶

Description.

## Types, macros and constants¶

fmpq_mat_struct
fmpq_mat_t

Description.

## Memory management¶

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.

void fmpq_mat_init_set(fmpq_mat_t mat1, const fmpq_mat_t mat2)

Initialises mat1 and sets it equal to mat2.

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.

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.

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¶

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.

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.

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.

slong fmpq_mat_nrows(const fmpq_mat_t mat)

Return the number of rows of the matrix mat.

slong fmpq_mat_ncols(const fmpq_mat_t mat)

Return the number of columns of the matrix mat.

## Basic assignment¶

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.

void fmpq_mat_zero(fmpq_mat_t mat)

Sets mat to the zero matrix.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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¶

void fmpq_mat_print(const fmpq_mat_t mat)

Prints the matrix mat to standard output.

## Random matrix generation¶

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.

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¶

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.

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¶

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 : $$(m + k) \times n$$.

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¶

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¶

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.

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.

int fmpq_mat_is_zero(const fmpq_mat_t mat)

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

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.

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.

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¶

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.

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.

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.

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.

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.

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.

void fmpq_mat_set_fmpz_mat(fmpq_mat_t dest, const fmpz_mat_t src)

Sets dest to src.

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¶

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.

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

Set 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¶

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.

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.

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.

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.

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.

## Kronecker product¶

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¶

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¶

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.

## Nonsingular solving¶

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.

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.

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.

## Inverse¶

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¶

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.

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.

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.

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¶

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¶

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¶

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.

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¶

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.

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.