Matrix Operations

Overview

Matrices in Giac are represented as vectors of vectors. The matrice type is an alias for vecteur, where each element is itself a vecteur representing a row.

Type Definition

typedef vecteur matrice;

A matrix is a vecteur where each element is a vecteur (row vector).

Matrix Properties

ckmatrix

Check if a structure is a valid matrix.

bool ckmatrix(const matrice & a);
bool ckmatrix(const matrice & a, bool allow_embedded_vect, bool ckundef = true);
bool ckmatrix(const gen & a);

const matrice &

Matrix to validate

allow_embedded_vect

bool

Allow nested vectors

ckundef

bool

default:"true"

Check for undefined values

return

bool

true if a is a valid matrix

is_squarematrix

Check if a matrix is square.

bool is_squarematrix(const matrice & a);
bool is_squarematrix(const gen & a);

const matrice &

Matrix to check

return

bool

true if matrix has equal rows and columns

Matrix Dimensions

int mrows(const matrice & a);
int mcols(const matrice & a);
void mdims(const matrice & m, int & r, int & c);

const matrice &

Input matrix

return

int

Number of rows (mrows) or columns (mcols)

Matrix Construction

midn

Create an identity matrix.

matrice midn(int n);
void midn(int n, matrice & res);

int

Size of the identity matrix

return

matrice

n×n identity matrix

Giac Command: _idn Example:

matrice I = midn(3); // 3×3 identity matrix

mranm

Generate a random matrix.

matrice mranm(int n, int m, const gen & f, GIAC_CONTEXT);

int

Number of rows

int

Number of columns

const gen &

Random element generator function

return

matrice

n×m random matrix

Giac Command: _ranm

makefreematrice

Create a matrix with modifiable rows.

matrice makefreematrice(const matrice & m);

const matrice &

Input matrix

return

matrice

Matrix with independent row vectors (safe for in-place modification)

Matrix Operations

mtran

Transpose a matrix.

matrice mtran(const matrice & a);
void mtran(const matrice & a, matrice & res, int ncolres = 0, bool ckundef = true);

const matrice &

Matrix to transpose

ncolres

int

default:"0"

Expected number of columns in result

return

matrice

Transposed matrix

Giac Command: _tran

mmult

Matrix multiplication.

matrice mmult(const matrice & a, const matrice & b);
void mmult(const matrice & a, const matrice & b, matrice & res);
bool mmultck(const matrice & a, const matrice & b, matrice & res);

const matrice &

First matrix

const matrice &

Second matrix

return

matrice

Product matrix a × b

mmult assumes dimensions are compatible. Use mmultck for dimension checking.

Matrix-Vector Operations

vecteur multmatvecteur(const matrice & a, const vecteur & b);
void multmatvecteur(const matrice & a, const vecteur & b, vecteur & res);

vecteur multvecteurmat(const vecteur & a, const matrice & b);
void multvecteurmat(const vecteur & a, const matrice & b, vecteur & res);

const matrice &

Matrix

const vecteur &

Vector

return

vecteur

Result of matrix-vector multiplication

mtrace

Compute matrix trace.

gen mtrace(const matrice & a);
gen ckmtrace(const gen & a, GIAC_CONTEXT);

const matrice &

Square matrix

return

gen

Sum of diagonal elements

Giac Command: _trace

Determinant and Inverse

mdet

Compute matrix determinant.

gen mdet(const matrice & a, GIAC_CONTEXT);

const matrice &

Square matrix

return

gen

Determinant of a

Giac Command: _det

minv

Compute matrix inverse.

matrice minv(const matrice & a, GIAC_CONTEXT);
bool minv(const matrice & a, matrice & res, bool convert_internal, int algorithm, GIAC_CONTEXT);

const matrice &

Square matrix to invert

convert_internal

bool

Convert to internal representation

algorithm

int

Algorithm selection (if unsure, use 1)

return

matrice

Inverse matrix a⁻¹

Returns empty matrix if a is singular.

Eigenvalues and Eigenvectors

megvl

Compute eigenvalues.

vecteur megvl(const matrice & a, GIAC_CONTEXT);

const matrice &

Square matrix

return

vecteur

Vector of eigenvalues

Giac Command: _egvl

megv

Compute eigenvectors.

matrice megv(const matrice & a, GIAC_CONTEXT);

const matrice &

Square matrix

return

matrice

Matrix of eigenvectors

Giac Command: _egv

egv

Full eigenvalue decomposition.

bool egv(const matrice & m, matrice & p, vecteur & d, GIAC_CONTEXT, 
         bool jordan, bool rational_jordan_form, bool eigenvalues_only);

const matrice &

Input square matrix

matrice &

Output eigenvector matrix

vecteur &

Output eigenvalues (or Jordan form if jordan=true)

jordan

bool

Return Jordan form instead of diagonal

rational_jordan_form

bool

Use rational Jordan form

eigenvalues_only

bool

Only compute eigenvalues

return

bool

true if successful

Characteristic Polynomial

mpcar

Compute characteristic polynomial.

vecteur mpcar(const matrice & a, vecteur & B, bool compute_B, GIAC_CONTEXT);
vecteur mpcar(const matrice & a, vecteur & Bv, bool compute_Bv, 
              bool convert_internal, GIAC_CONTEXT);

const matrice &

Square matrix

vecteur &

Output parameter for additional computation

compute_B

bool

Whether to compute B parameter

return

vecteur

Coefficients of characteristic polynomial

Giac Command: _pcar

Kernel and Image

mker

Compute kernel (null space) of a matrix.

vecteur mker(const matrice & a, GIAC_CONTEXT);
bool mker(const matrice & a, vecteur & v, int algorithm, GIAC_CONTEXT);

const matrice &

Input matrix

algorithm

int

default:"0"

Algorithm selection

return

vecteur

Basis vectors for the kernel

Giac Command: _ker

mimage

Compute image (column space) of a matrix.

vecteur mimage(const matrice & a, GIAC_CONTEXT);
bool mimage(const matrice & a, vecteur & v, GIAC_CONTEXT);

const matrice &

Input matrix

return

vecteur

Basis vectors for the column space

Giac Command: _image

Matrix Decompositions

mlu

LU decomposition.

bool mlu(const matrice & a0, vecteur & P, matrice & L, matrice & U, GIAC_CONTEXT);

const matrice &

Input matrix

vecteur &

Permutation vector

matrice &

Lower triangular matrix

matrice &

Upper triangular matrix

return

bool

true if decomposition successful

Giac Command: _lu

QR Decomposition

Giac Command: _qr

Cholesky Decomposition

Giac Command: _cholesky

SVD (Singular Value Decomposition)

Giac Command: _svd

LDL Decomposition

bool ldl(matrice & a, std::vector<int> & perm, int mat_type, bool & sing, 
         double time_limit, GIAC_CONTEXT);

matrice &

Input/output matrix

perm

std::vector<int> &

Permutation vector

mat_type

int

Matrix type indicator

sing

bool &

Set to true if matrix is singular

time_limit

double

Maximum computation time

return

bool

true if decomposition successful

Giac Command: _ldl

Jordan Form

mjordan

Compute Jordan normal form.

vecteur mjordan(const matrice & a, bool rational_jordan, GIAC_CONTEXT);
gen jordan(const gen & a, bool rational_jordan, GIAC_CONTEXT);

const matrice &

Square matrix

rational_jordan

bool

Use rational Jordan form

return

vecteur

Jordan normal form decomposition

Giac Commands: _jordan, _rat_jordan

Special Matrix Operations

matpow

Matrix power.

matrice matpow(const matrice & m, const gen & n, GIAC_CONTEXT);

const matrice &

Base matrix

const gen &

Exponent

return

matrice

Matrix power mⁿ

Giac Command: _matpow

Hessenberg Form

void hessenberg(std_matrix<gen> & H, std_matrix<gen> & P, GIAC_CONTEXT);
void hessenberg_ortho(std_matrix<gen> & H, std_matrix<gen> & P, GIAC_CONTEXT);

Giac Command: _hessenberg

Gram-Schmidt Orthogonalization

matrice gramschmidt(const matrice & m, bool normalize, GIAC_CONTEXT);
matrice gramschmidt(const matrice & m, matrice & r, bool normalize, GIAC_CONTEXT);

const matrice &

Input matrix with column vectors

normalize

bool

Normalize the output vectors

matrice &

R matrix from QR decomposition

return

matrice

Orthogonalized matrix

LLL Algorithm

matrice lll(const matrice & m, GIAC_CONTEXT);
matrice lll(const matrice & M, matrice & L, matrice & O, matrice & A, GIAC_CONTEXT);

const matrice &

Input lattice basis

return

matrice

LLL-reduced basis

Giac Command: _lll

Integer Matrix Operations

Smith Normal Form

bool ismith(const matrice & Aorig, matrice & U, matrice & A, matrice & V, GIAC_CONTEXT);

Giac Command: _ismith

Hermite Normal Form

bool ihermite(const matrice & Aorig, matrice & U, matrice & A, GIAC_CONTEXT);

Giac Command: _ihermite

Core API

Algebra

Calculus

Linear Algebra

Solvers

Utilities

​Overview

​Type Definition

​Matrix Properties

​ckmatrix

​is_squarematrix

​Matrix Dimensions

​Matrix Construction

​midn

​mranm

​makefreematrice

​Matrix Operations

​mtran

​mmult

​Matrix-Vector Operations

​mtrace

​Determinant and Inverse

​mdet

​minv

​Eigenvalues and Eigenvectors

​megvl

​megv

​egv

​Characteristic Polynomial

​mpcar

​Kernel and Image

​mker

​mimage

​Matrix Decompositions

​mlu

​QR Decomposition

​Cholesky Decomposition

​SVD (Singular Value Decomposition)

​LDL Decomposition

​Jordan Form

​mjordan

​Special Matrix Operations

​matpow

​Hessenberg Form

​Gram-Schmidt Orthogonalization

​LLL Algorithm

​Integer Matrix Operations

​Smith Normal Form

​Hermite Normal Form

​See Also

Build docs developers (and LLMs) love

Overview

Type Definition

Matrix Properties

ckmatrix

is_squarematrix

Matrix Dimensions

Matrix Construction

midn

mranm

makefreematrice

Matrix Operations

mtran

mmult

Matrix-Vector Operations

mtrace

Determinant and Inverse

mdet

minv

Eigenvalues and Eigenvectors

megvl

megv

egv

Characteristic Polynomial

mpcar

Kernel and Image

mker

mimage

Matrix Decompositions

mlu

QR Decomposition

Cholesky Decomposition

SVD (Singular Value Decomposition)

LDL Decomposition

Jordan Form

mjordan

Special Matrix Operations

matpow

Hessenberg Form

Gram-Schmidt Orthogonalization

LLL Algorithm

Integer Matrix Operations

Smith Normal Form

Hermite Normal Form

See Also