# nmod_mat.h – matrices over integers mod n (word-size n)¶

Description.

## Types, macros and constants¶

nmod_mat_struct
nmod_mat_t

Description.

## Memory management¶

void nmod_mat_init(nmod_mat_t mat, slong rows, slong cols, mp_limb_t n)

Initialises mat to a rows-by-cols matrix with coefficients modulo~n, where $$n$$ can be any nonzero integer that fits in a limb. All elements are set to zero.

void nmod_mat_init_set(nmod_mat_t mat, nmod_mat_t src)

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

void nmod_mat_clear(nmod_mat_t mat)

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 nmod_mat_t object.

void nmod_mat_set(nmod_mat_t mat, nmod_mat_t src)

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

void nmod_mat_swap(nmod_mat_t mat1, nmod_mat_t mat2)

Exchanges mat1 and mat2.

void nmod_mat_swap_entrywise(nmod_mat_t mat1, nmod_mat_t mat2)

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

## Basic properties and manipulation¶

MACRO nmod_mat_entry(nmod_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. This macro can be used both for reading and writing coefficients.

mp_limb_t nmod_mat_get_entry(const nmod_mat_t mat, slong i, slong j)

Get the entry at row $$i$$ and column $$j$$ of the matrix mat.

mp_limb_t * nmod_mat_entry_ptr(const nmod_mat_t mat, slong i, slong j)

Return a pointer to the entry at row $$i$$ and column $$j$$ of the matrix mat.

void nmod_mat_set_entry(nmod_mat_t mat, slong i, slong j, mp_limb_t x)

Set the entry at row $$i$$ and column $$j$$ of the matrix mat to x.

slong nmod_mat_nrows(nmod_mat_t mat)

Returns the number of rows in mat.

slong nmod_mat_ncols(nmod_mat_t mat)

Returns the number of columns in mat.

void nmod_mat_zero(nmod_mat_t mat)

Sets all entries of the matrix mat to zero.

int nmod_mat_is_zero(nmod_mat_t mat)

Returns $$1$$ if all entries of the matrix mat are zero.

## Window¶

void nmod_mat_window_init(nmod_mat_t window, const nmod_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 nmod_mat_window_clear(nmod_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 nmod_mat_concat_vertical(nmod_mat_t res, const nmod_mat_t mat1, const nmod_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 nmod_mat_concat_horizontal(nmod_mat_t res, const nmod_mat_t mat1, const nmod_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 : $$m \times (n + k)$$.

## Printing¶

void nmod_mat_print_pretty(nmod_mat_t mat)

Pretty-prints mat to stdout. A header is printed followed by the rows enclosed in brackets. Each column is right-aligned to the width of the modulus written in decimal, and the columns are separated by spaces. For example:

<2 x 3 integer matrix mod 2903>
[   0    0 2607]
[ 622    0    0]


## Random matrix generation¶

void nmod_mat_randtest(nmod_mat_t mat, flint_rand_t state)

Sets the elements to a random matrix with entries between $$0$$ and $$m-1$$ inclusive, where $$m$$ is the modulus of mat. A sparse matrix is generated with increased probability.

void nmod_mat_randfull(nmod_mat_t mat, flint_rand_t state)

Sets the element to random numbers likely to be close to the modulus of the matrix. This is used to test potential overflow-related bugs.

int nmod_mat_randpermdiag(nmod_mat_t mat, mp_limb_t * diag, slong n, flint_rand_t state)

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 nmod_mat_randrank(nmod_mat_t mat, slong rank, flint_rand_t state)

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 integers between $$0$$ and $$m-1$$ inclusive, where $$m$$ is the modulus of mat.

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

void nmod_mat_randops(nmod_mat_t mat, slong count, flint_rand_t state)

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 nmod_mat_randtril(nmod_mat_t mat, flint_rand_t state, int unit)

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 nmod_mat_randtriu(nmod_mat_t mat, flint_rand_t state, int unit)

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 nmod_mat_equal(nmod_mat_t mat1, nmod_mat_t mat2)

Returns nonzero if mat1 and mat2 have the same dimensions and elements, and zero otherwise. The moduli are ignored.

int nmod_mat_is_zero_row(const nmod_mat_t mat, slong i)

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

## Transpose¶

void nmod_mat_transpose(nmod_mat_t B, nmod_mat_t A)

Sets $$B$$ to the transpose of $$A$$. Dimensions must be compatible. $$B$$ and $$A$$ may be the same object if and only if the matrix is square.

void nmod_mat_swap_rows(nmod_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 nmod_mat_swap_cols(nmod_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 nmod_mat_invert_rows(nmod_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 nmod_mat_invert_cols(nmod_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 nmod_mat_add(nmod_mat_t C, nmod_mat_t A, nmod_mat_t B)

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

void nmod_mat_sub(nmod_mat_t C, nmod_mat_t A, nmod_mat_t B)

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

void nmod_mat_neg(nmod_mat_t A, nmod_mat_t B)

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

## Matrix-scalar arithmetic¶

void nmod_mat_scalar_mul(nmod_mat_t B, const nmod_mat_t A, mp_limb_t c)

Sets $$B = cA$$, where the scalar $$c$$ is assumed to be reduced modulo the modulus. Dimensions of $$A$$ and $$B$$ must be identical.

void nmod_mat_scalar_addmul_ui(nmod_mat_t dest, const nmod_mat_t X, const nmod_mat_t Y, const mp_limb_t b)

Sets $$dest = X + bY$$, where the scalar $$b$$ is assumed to be reduced modulo the modulus. Dimensions of dest, X and Y must be identical. dest can be aliased with X or Y.

void nmod_mat_scalar_mul_fmpz(nmod_mat_t res, const nmod_mat_t M, const fmpz_t c)

Sets $$B = cA$$, where the scalar $$c$$ is of type fmpz_t. Dimensions of $$A$$ and $$B$$ must be identical.

## Matrix multiplication¶

void nmod_mat_mul(nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B)

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

void _nmod_mat_mul_classical_op(nmod_mat_t D, const nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B, int op)

Sets D = A*B op C where op is +1 for addition, -1 for subtraction and 0 to ignore C.

void nmod_mat_mul_classical(nmod_mat_t C, const nmod_mat_t A, const nmod_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 classical matrix multiplication, creating a temporary transposed copy of $$B$$ to improve memory locality if the matrices are large enough, and packing several entries of $$B$$ into each word if the modulus is very small.

void _nmod_mat_mul_classical_threaded_pool_op(nmod_mat_t D, const nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B, int op, thread_pool_handle * threads, slong num_threads)

Multithreaded version of _nmod_mat_mul_classical.

void _nmod_mat_mul_classical_threaded_op(nmod_mat_t D, const nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B, int op)

Multithreaded version of _nmod_mat_mul_classical.

void nmod_mat_mul_classical_threaded(nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B)

Multithreaded version of nmod_mat_mul_classical.

void nmod_mat_mul_strassen(nmod_mat_t C, const nmod_mat_t A, const nmod_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).

int nmod_mat_mul_blas(nmod_mat_t C, const nmod_mat_t A, const nmod_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.

void nmod_mat_addmul(nmod_mat_t D, const nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B)

Sets $$D = C + AB$$. $$C$$ and $$D$$ may be aliased with each other but not with $$A$$ or $$B$$. Automatically selects between classical and Strassen multiplication.

void nmod_mat_submul(nmod_mat_t D, const nmod_mat_t C, const nmod_mat_t A, const nmod_mat_t B)

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

void nmod_mat_mul_nmod_vec(mp_limb_t * c, const nmod_mat_t A, const mp_limb_t * b, slong blen)
void nmod_mat_mul_nmod_vec_ptr(mp_limb_t * const * c, const nmod_mat_t A, const mp_limb_t * 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 nmod_mat_nmod_vec_mul(mp_limb_t * c, const mp_limb_t * a, slong alen, const nmod_mat_t B)
void nmod_mat_nmod_vec_mul_ptr(mp_limb_t * const * c, const mp_limb_t * const * a, slong alen, const nmod_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.

## Matrix Exponentiation¶

void _nmod_mat_pow(nmod_mat_t dest, const nmod_mat_t mat, ulong pow)

Sets $$dest = mat^{pow}$$. dest and mat cannot be aliased. Implements exponentiation by squaring.

void nmod_mat_pow(nmod_mat_t dest, nmod_mat_t mat, ulong pow)
Sets $$dest = mat^{pow}$$. dest and mat may be aliased. Implements
exponentiation by squaring.

## Trace¶

mp_limb_t nmod_mat_trace(const nmod_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 and rank¶

mp_limb_t nmod_mat_det_howell(const nmod_mat_t A)

Returns the determinant of $$A$$.

mp_limb_t nmod_mat_det(const nmod_mat_t A)

Returns the determinant of $$A$$.

slong nmod_mat_rank(const nmod_mat_t A)

Returns the rank of $$A$$. The modulus of $$A$$ must be a prime number.

## Inverse¶

int nmod_mat_inv(nmod_mat_t B, nmod_mat_t A)

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 and modulus. The modulus must be prime.

## Triangular solving¶

void nmod_mat_solve_tril(nmod_mat_t X, const nmod_mat_t L, const nmod_mat_t B, int unit)

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 nmod_mat_solve_tril_classical(nmod_mat_t X, const nmod_mat_t L, const nmod_mat_t B, int unit)

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 nmod_mat_solve_tril_recursive(nmod_mat_t X, const nmod_mat_t L, const nmod_mat_t B, int unit)

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 nmod_mat_solve_triu(nmod_mat_t X, const nmod_mat_t U, const nmod_mat_t B, int unit)

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 nmod_mat_solve_triu_classical(nmod_mat_t X, const nmod_mat_t U, const nmod_mat_t B, int unit)

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 nmod_mat_solve_triu_recursive(nmod_mat_t X, const nmod_mat_t U, const nmod_mat_t B, int unit)

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.

## Nonsingular square solving¶

int nmod_mat_solve(nmod_mat_t X, nmod_mat_t A, nmod_mat_t B)

Solves the matrix-matrix equation $$AX = B$$ over $$\mathbb{Z} / p \mathbb{Z}$$ where $$p$$ is the modulus of $$X$$ which must be a prime number. $$X$$, $$A$$, and $$B$$ should have the same moduli.

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 nmod_mat_can_solve_inner(slong * rank, slong * perm, slong * pivots, nmod_mat_t X, const nmod_mat_t A, const nmod_mat_t B)

As for nmod_mat_can_solve() except that if $$rank$$ is not $$NULL$$ the value it points to will be set to the rank of $$A$$. If $$perm$$ is not $$NULL$$ then it must be a valid initialised permutation whose length is the number of rows of $$A$$. After the function call it will be set to the row permutation given by LU decomposition of $$A$$. If $$pivots$$ is not $$NULL$$ then it must an initialised vector. Only the first $$*rank$$ of these will be set by the function call. They are set to the columns of the pivots chosen by the LU decomposition of $$A$$.

int nmod_mat_can_solve(nmod_mat_t X, nmod_mat_t A, nmod_mat_t B)

Solves the matrix-matrix equation $$AX = B$$ over $$\mathbb{Z} / p \mathbb{Z}$$ where $$p$$ is the modulus of $$X$$ which must be a prime number. $$X$$, $$A$$, and $$B$$ should have the same moduli.

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.

int nmod_mat_solve_vec(mp_limb_t * x, nmod_mat_t A, mp_limb_t * b)

Solves the matrix-vector equation $$Ax = b$$ over $$\mathbb{Z} / p \mathbb{Z}$$ where $$p$$ is the modulus of $$A$$ which must be a prime number.

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

## LU decomposition¶

slong nmod_mat_lu(slong * P, nmod_mat_t A, int rank_check)

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 nmod_mat_lu_recursive().

slong nmod_mat_lu_classical(slong * P, nmod_mat_t A, int rank_check)

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 nmod_mat_lu(). Uses Gaussian elimination.

slong nmod_mat_lu_recursive(slong * P, nmod_mat_t A, int rank_check)

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 nmod_mat_lu(). Uses recursive block decomposition, switching to classical Gaussian elimination for sufficiently small blocks.

## Reduced row echelon form¶

slong nmod_mat_rref(nmod_mat_t A)

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 nmod_mat_reduce_row(nmod_mat_t A, slong * P, slong * L, slong n)

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.

## Nullspace¶

slong nmod_mat_nullspace(nmod_mat_t X, const nmod_mat_t A)

Computes the nullspace of $$A$$ and returns the nullity.

More precisely, this function sets $$X$$ to a maximum rank matrix such that $$AX = 0$$ and returns the rank of $$X$$. The columns of $$X$$ will form a basis for the nullspace of $$A$$.

$$X$$ must have sufficient space to store all basis vectors in the nullspace.

This function computes the reduced row echelon form and then reads off the basis vectors.

## Transforms¶

void nmod_mat_similarity(nmod_mat_t M, slong r, ulong 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.

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

## Characteristic polynomial¶

void nmod_mat_charpoly_berkowitz(nmod_poly_t p, const nmod_mat_t M)
void nmod_mat_charpoly_danilevsky(nmod_poly_t p, const nmod_mat_t M)
void nmod_mat_charpoly(nmod_poly_t p, const nmod_mat_t M)

Compute the characteristic polynomial $$p$$ of the matrix $$M$$. The matrix is required to be square, otherwise an exception is raised. The danilevsky algorithm assumes that the modulus is prime.

## Minimal polynomial¶

void nmod_mat_minpoly(nmod_poly_t p, const nmod_mat_t M)

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

## Strong echelon form and Howell form¶

void nmod_mat_strong_echelon_form(nmod_mat_t A)

Puts $$A$$ into strong echelon form. 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. Note that [FieHof2014] defines strong echelon form as a lower left normal form, while the implemented version returns an upper right normal form, agreeing with the definition of Howell form in [StoMul1998].

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

slong nmod_mat_howell_form`(nmod_mat_t A)

Puts $$A$$ into Howell form and returns the number of non-zero rows. 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.