# fq_mat.h – matrices over finite fields¶

Description.

## Types, macros and constants¶

fq_mat_struct
fq_mat_t

Description.

## Memory management¶

void fq_mat_init(fq_mat_t mat, slong rows, slong cols, const fq_ctx_t ctx)

Initialises mat to a rows-by-cols matrix with coefficients in $$\mathbf{F}_{q}$$ given by ctx. All elements are set to zero.

void fq_mat_init_set(fq_mat_t mat, fq_mat_t src, const fq_ctx_t ctx)

Initialises mat and sets its dimensions and elements to those of src.

void fq_mat_clear(fq_mat_t mat, const fq_ctx_t ctx)

Clears the matrix and releases any memory it used. The matrix cannot be used again until it is initialised. This function must be called exactly once when finished using an fq_mat_t object.

void fq_mat_set(fq_mat_t mat, fq_mat_t src, const fq_ctx_t ctx)

Sets mat to a copy of src. It is assumed that mat and src have identical dimensions.

## Basic properties and manipulation¶

fq_struct * fq_mat_entry(fq_mat_t mat, slong i, slong j)

Directly accesses the entry in mat in row $$i$$ and column $$j$$, indexed from zero. No bounds checking is performed.

void fq_mat_entry_set(fq_mat_t mat, slong i, slong j, fq_t x, const fq_ctx_t ctx)

Sets the entry in mat in row $$i$$ and column $$j$$ to x.

slong fq_mat_nrows(fq_mat_t mat, const fq_ctx_t ctx)

Returns the number of rows in mat.

slong fq_mat_ncols(fq_mat_t mat, const fq_ctx_t ctx)

Returns the number of columns in mat.

void fq_mat_swap(fq_mat_t mat1, fq_mat_t mat2, const fq_ctx_t ctx)

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

void fq_mat_swap_entrywise(fq_mat_t mat1, fq_mat_t mat2)

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

void fq_mat_zero(fq_mat_t mat, const fq_ctx_t ctx)

Sets all entries of mat to 0.

void fq_mat_one(fq_mat_t mat, const fq_ctx_t ctx)

Sets all the diagonal entries of mat to 1 and all other entries to 0.

void fq_mat_swap_rows(fq_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 fq_mat_swap_cols(fq_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 fq_mat_invert_rows(fq_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 fq_mat_invert_cols(fq_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.

## Conversions¶

void fq_mat_set_nmod_mat(fq_mat_t mat1, const nmod_mat_t mat2, const fq_ctx_t ctx)

Sets the matrix mat1 to the matrix mat2.

void fq_mat_set_fmpz_mod_mat(fq_mat_t mat1, const fmpz_mod_mat_t mat2, const fq_ctx_t ctx)

Sets the matrix mat1 to the matrix mat2.

## Concatenate¶

void fq_mat_concat_vertical(fq_mat_t res, const fq_mat_t mat1, const fq_mat_t mat2, const fq_ctx_t ctx)

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 fq_mat_concat_horizontal(fq_mat_t res, const fq_mat_t mat1, const fq_mat_t mat2, const fq_ctx_t ctx)

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

## Printing¶

void fq_mat_print_pretty(const fq_mat_t mat, const fq_ctx_t ctx)

Pretty-prints mat to stdout. A header is printed followed by the rows enclosed in brackets.

int fq_mat_fprint_pretty(FILE * file, const fq_mat_t mat, const fq_ctx_t ctx)

Pretty-prints mat to file. A header is printed followed by the rows enclosed in brackets.

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

void fq_mat_print(const fq_mat_t mat, const fq_ctx_t ctx)

Prints mat to stdout. A header is printed followed by the rows enclosed in brackets.

int fq_mat_fprint(FILE * file, const fq_mat_t mat, const fq_ctx_t ctx)

Prints mat to file. A header is printed followed by the rows enclosed in brackets.

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

## Window¶

void fq_mat_window_init(fq_mat_t window, const fq_mat_t mat, slong r1, slong c1, slong r2, slong c2, const fq_ctx_t ctx)

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 fq_mat_window_clear(fq_mat_t window, const fq_ctx_t ctx)

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¶

void fq_mat_randtest(fq_mat_t mat, flint_rand_t state, const fq_ctx_t ctx)

Sets the elements of mat to random elements of $$\mathbf{F}_{q}$$, given by ctx.

int fq_mat_randpermdiag(fq_mat_t mat, fq_struct * diag, slong n, flint_rand_t state, const fq_ctx_t ctx)

Sets mat to a random permutation of the diagonal matrix with $$n$$ leading entries given by the vector diag. It is assumed that the main diagonal of mat has room for at least $$n$$ entries.

Returns $$0$$ or $$1$$, depending on whether the permutation is even or odd respectively.

void fq_mat_randrank(fq_mat_t mat, slong rank, flint_rand_t state, const fq_ctx_t ctx)

Sets mat to a random sparse matrix with the given rank, having exactly as many non-zero elements as the rank, with the non-zero elements being uniformly random elements of $$\mathbf{F}_{q}$$.

The matrix can be transformed into a dense matrix with unchanged rank by subsequently calling fq_mat_randops().

void fq_mat_randops(fq_mat_t mat, slong count, flint_rand_t state, const fq_ctx_t ctx)

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, determinant) unchanged.

void fq_mat_randtril(fq_mat_t mat, flint_rand_t state, int unit, const fq_ctx_t ctx)

Sets mat to a random lower triangular matrix. If unit is 1, it will have ones on the main diagonal, otherwise it will have random nonzero entries on the main diagonal.

void fq_mat_randtriu(fq_mat_t mat, flint_rand_t state, int unit, const fq_ctx_t ctx)

Sets mat to a random upper triangular matrix. If unit is 1, it will have ones on the main diagonal, otherwise it will have random nonzero entries on the main diagonal.

## Comparison¶

int fq_mat_equal(fq_mat_t mat1, fq_mat_t mat2, const fq_ctx_t ctx)

Returns nonzero if mat1 and mat2 have the same dimensions and elements, and zero otherwise.

int fq_mat_is_zero(const fq_mat_t mat, const fq_ctx_t ctx)

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

int fq_mat_is_one(const fq_mat_t mat, const fq_ctx_t ctx)

Returns a non-zero value if all entries mat are zero except the diagonal entries which must be one, otherwise returns zero..

int fq_mat_is_empty(const fq_mat_t mat, const fq_ctx_t ctx)

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

int fq_mat_is_square(const fq_mat_t mat, const fq_ctx_t ctx)

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

void fq_mat_add(fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Computes $$C = A + B$$. Dimensions must be identical.

void fq_mat_sub(fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Computes $$C = A - B$$. Dimensions must be identical.

void fq_mat_neg(fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Sets $$B = -A$$. Dimensions must be identical.

## Matrix multiplication¶

void fq_mat_mul(fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Sets $$C = AB$$. Dimensions must be compatible for matrix multiplication. Aliasing is allowed. This function automatically chooses between classical and KS multiplication.

void fq_mat_mul_classical(fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Sets $$C = AB$$. Dimensions must be compatible for matrix multiplication. $$C$$ is not allowed to be aliased with $$A$$ or $$B$$. Uses classical matrix multiplication.

void fq_mat_mul_KS(fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Sets $$C = AB$$. Dimensions must be compatible for matrix multiplication. $$C$$ is not allowed to be aliased with $$A$$ or $$B$$. Uses Kronecker substitution to perform the multiplication over the integers.

void fq_mat_submul(fq_mat_t D, const fq_mat_t C, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Sets $$D = C + AB$$. $$C$$ and $$D$$ may be aliased with each other but not with $$A$$ or $$B$$.

void fq_mat_mul_vec(fq_struct * c, const fq_mat_t A, const fq_struct * b, slong blen)
void fq_mat_mul_vec_ptr(fq_struct * const * c, const fq_mat_t A, const fq_struct * 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.

void fq_mat_vec_mul(fq_struct * c, const fq_struct * a, slong alen, const fq_mat_t B)
void fq_mat_vec_mul_ptr(fq_struct * const * c, const fq_struct * const * a, slong alen, const fq_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.

## Inverse¶

int fq_mat_inv(fq_mat_t B, fq_mat_t A, fq_ctx_t ctx)

Sets $$B = A^{-1}$$ and returns $$1$$ if $$A$$ is invertible. If $$A$$ is singular, returns $$0$$ and sets the elements of $$B$$ to undefined values.

$$A$$ and $$B$$ must be square matrices with the same dimensions.

## LU decomposition¶

slong fq_mat_lu(slong * P, fq_mat_t A, int rank_check, const fq_ctx_t ctx)

Computes a generalised LU decomposition $$LU = PA$$ of a given matrix $$A$$, returning the rank of $$A$$.

If $$A$$ is a nonsingular square matrix, it will be overwritten with a unit diagonal lower triangular matrix $$L$$ and an upper triangular matrix $$U$$ (the diagonal of $$L$$ will not be stored explicitly).

If $$A$$ is an arbitrary matrix of rank $$r$$, $$U$$ will be in row echelon form having $$r$$ nonzero rows, and $$L$$ will be lower triangular but truncated to $$r$$ columns, having implicit ones on the $$r$$ first entries of the main diagonal. All other entries will be zero.

If a nonzero value for rank_check is passed, the function will abandon the output matrix in an undefined state and return 0 if $$A$$ is detected to be rank-deficient.

This function calls fq_mat_lu_recursive.

slong fq_mat_lu_classical(slong * P, fq_mat_t A, int rank_check, const fq_ctx_t ctx)

Computes a generalised LU decomposition $$LU = PA$$ of a given matrix $$A$$, returning the rank of $$A$$. The behavior of this function is identical to that of fq_mat_lu. Uses Gaussian elimination.

slong fq_mat_lu_recursive(slong * P, fq_mat_t A, int rank_check, const fq_ctx_t ctx)

Computes a generalised LU decomposition $$LU = PA$$ of a given matrix $$A$$, returning the rank of $$A$$. The behavior of this function is identical to that of fq_mat_lu. Uses recursive block decomposition, switching to classical Gaussian elimination for sufficiently small blocks.

## Reduced row echelon form¶

slong fq_mat_rref(fq_mat_t A, const fq_ctx_t ctx)

Puts $$A$$ in reduced row echelon form and returns the rank of $$A$$.

The rref is computed by first obtaining an unreduced row echelon form via LU decomposition and then solving an additional triangular system.

slong fq_mat_reduce_row(fq_mat_t A, slong * P, slong * L, slong n, const fq_ctx_t ctx)

Reduce row n of the matrix $$A$$, assuming the prior rows are in Gauss form. However those rows may not be in order. The entry $$i$$ of the array $$P$$ is the row of $$A$$ which has a pivot in the $$i$$-th column. If no such row exists, the entry of $$P$$ will be $$-1$$. The function returns the column in which the $$n$$-th row has a pivot after reduction. This will always be chosen to be the first available column for a pivot from the left. This information is also updated in $$P$$. Entry $$i$$ of the array $$L$$ contains the number of possibly nonzero columns of $$A$$ row $$i$$. This speeds up reduction in the case that $$A$$ is chambered on the right. Otherwise the entries of $$L$$ can all be set to the number of columns of $$A$$. We require the entries of $$L$$ to be monotonic increasing.

## Triangular solving¶

void fq_mat_solve_tril(fq_mat_t X, const fq_mat_t L, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = L^{-1} B$$ where $$L$$ is a full rank lower triangular square matrix. If unit = 1, $$L$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed. Automatically chooses between the classical and recursive algorithms.

void fq_mat_solve_tril_classical(fq_mat_t X, const fq_mat_t L, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = L^{-1} B$$ where $$L$$ is a full rank lower triangular square matrix. If unit = 1, $$L$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed. Uses forward substitution.

void fq_mat_solve_tril_recursive(fq_mat_t X, const fq_mat_t L, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = L^{-1} B$$ where $$L$$ is a full rank lower triangular square matrix. If unit = 1, $$L$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed.

Uses the block inversion formula

$\begin{split}\begin{pmatrix} A & 0 \\ C & D \end{pmatrix}^{-1} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} A^{-1} X \\ D^{-1} ( Y - C A^{-1} X ) \end{pmatrix}\end{split}$

to reduce the problem to matrix multiplication and triangular solving of smaller systems.

void fq_mat_solve_triu(fq_mat_t X, const fq_mat_t U, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = U^{-1} B$$ where $$U$$ is a full rank upper triangular square matrix. If unit = 1, $$U$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed. Automatically chooses between the classical and recursive algorithms.

void fq_mat_solve_triu_classical(fq_mat_t X, const fq_mat_t U, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = U^{-1} B$$ where $$U$$ is a full rank upper triangular square matrix. If unit = 1, $$U$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed. Uses forward substitution.

void fq_mat_solve_triu_recursive(fq_mat_t X, const fq_mat_t U, const fq_mat_t B, int unit, const fq_ctx_t ctx)

Sets $$X = U^{-1} B$$ where $$U$$ is a full rank upper triangular square matrix. If unit = 1, $$U$$ is assumed to have ones on its main diagonal, and the main diagonal will not be read. $$X$$ and $$B$$ are allowed to be the same matrix, but no other aliasing is allowed.

Uses the block inversion formula

$\begin{split}\begin{pmatrix} A & B \\ 0 & D \end{pmatrix}^{-1} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} A^{-1} (X - B D^{-1} Y) \\ D^{-1} Y \end{pmatrix}\end{split}$

to reduce the problem to matrix multiplication and triangular solving of smaller systems.

## Solving¶

int fq_mat_solve(fq_mat_t X, const fq_mat_t A, const fq_mat_t B, const fq_ctx_t ctx)

Solves the matrix-matrix equation $$AX = B$$.

Returns $$1$$ if $$A$$ has full rank; otherwise returns $$0$$ and sets the elements of $$X$$ to undefined values.

The matrix $$A$$ must be square.

int fq_mat_can_solve(fq_mat_t X, fq_mat_t A, fq_mat_t B, const fq_ctx_t ctx)

Solves the matrix-matrix equation $$AX = B$$ over $$Fq$$.

Returns $$1$$ if a solution exists; otherwise returns $$0$$ and sets the elements of $$X$$ to zero. If more than one solution exists, one of the valid solutions is given.

There are no restrictions on the shape of $$A$$ and it may be singular.

## Transforms¶

void fq_mat_similarity(fq_mat_t M, slong r, fq_t d, fq_ctx_t ctx)

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.

The value $$d$$ is required to be reduced modulo the modulus of the entries in the matrix.

## Characteristic polynomial¶

void fq_mat_charpoly_danilevsky(fq_poly_t p, const fq_mat_t M, const fq_ctx_t ctx)

Compute the characteristic polynomial $$p$$ of the matrix $$M$$. The matrix is assumed to be square.

void fq_mat_charpoly(fq_poly_t p, const fq_mat_t M, const fq_ctx_t ctx)

Compute the characteristic polynomial $$p$$ of the matrix $$M$$. The matrix is required to be square, otherwise an exception is raised.

## Minimal polynomial¶

void fq_mat_minpoly(fq_poly_t p, const fq_mat_t M, const fq_ctx_t ctx)

Compute the minimal polynomial $$p$$ of the matrix $$M$$. The matrix is required to be square, otherwise an exception is raised.