A templated matrix class supporting various linear algebra operations. More...

#include <matrix.h>

Public Member Functions
	Matrix ()
	Default constructor creating an empty 0×0 matrix.

	Matrix (int r, int c)
	Constructs a matrix with specified dimensions, initialized with default values.

	Matrix (const std::vector< std::vector< T > > &mat)
	Constructs a matrix from a 2D vector.

T &	operator() (int row, int column)
	Accesses matrix element at specified position (non-const).

const T &	operator() (int row, int column) const
	Accesses matrix element at specified position (const).

std::vector< T > &	operator[] (int i)
	Accesses a row of the matrix (non-const).

const std::vector< T > &	operator[] (int i) const
	Accesses a row of the matrix (const).

Matrix< T >	operator+ (const Matrix< T > &other) const
	Matrix addition.

Matrix< T >	operator- (const Matrix< T > &other) const
	Matrix subtraction.

Matrix< T >	operator* (const Matrix< T > &other) const
	Matrix multiplication.

Matrix< T >	operator* (const T &scalar) const
	Scalar multiplication.

Matrix< T >	operator/= (const Matrix< T > &other) const
	Element-wise division.

Matrix< T >	operator-= (const Matrix< T > &other) const
	In-place subtraction.

T	determinant () const
	Computes the determinant of the matrix.

Matrix< T >	inverse () const
	Computes the inverse of the matrix.

Matrix< T >	transpose () const
	Computes the transpose of the matrix.

std::vector< T >	eigenvalues () const
	Computes the eigenvalues of the matrix.

Matrix< T >	eigenvectors () const
	Computes the eigenvectors of the matrix.

Matrix< T >	adjoint () const
	Computes the adjoint (adjugate) matrix.

Matrix< T >	submatrix (int row_start, int row_end, int col_start, int col_end) const
	Extracts a submatrix from specified row and column ranges.

T	rank () const
	Computes the rank of the matrix.

bool	isSymmetric () const
	Checks if the matrix is symmetric.

std::pair< Matrix< T >, Matrix< T > >	LUDecomposition () const
	Performs LU decomposition of the matrix.

std::pair< Matrix< T >, Matrix< T > >	QRDecomposition () const
	Performs QR decomposition of the matrix using Gram-Schmidt orthogonalization.

std::vector< T >	matVec (const std::vector< T > &v) const
	Performs matrix-vector multiplication.

T	dot (const std::vector< T > &a, const std::vector< T > &b) const
	Computes the dot product of two vectors.

void	normalize (std::vector< T > &v) const
	Normalizes a vector to unit length (in-place).

void	lanczos (int m, std::vector< T > &alpha, std::vector< T > &beta, Matrix< T > &Q) const
	Performs the Lanczos algorithm for tridiagonalization.

Matrix< T >	buildTridiagonal (const std::vector< T > &alpha, const std::vector< T > &beta, int m) const
	Constructs a tridiagonal matrix from diagonal and off-diagonal elements.

double	frobeniusNorm () const
	Computes the Frobenius norm of the matrix.

double	spectralNorm () const
	Computes the spectral norm (2-norm) of the matrix.

double	norm (const std::vector< T > &v) const
	Computes the Euclidean (L2) norm of a vector.

Matrix< T >	getMinor (int row, int col) const
	Gets the minor matrix by removing a specified row and column.

std::string	toString () const
	Converts the matrix to a formatted string representation.

int	getRows () const
	Gets the number of rows in the matrix.

int	getCols () const
	Gets the number of columns in the matrix.

Static Public Member Functions
static Matrix< T >	identity (int n)
	Creates an n × n identity matrix.

static Matrix< T >	zeros (int r, int c)
	Creates an r × c matrix filled with zeros.

static Matrix< T >	ones (int r, int c)
	Creates an r × c matrix filled with ones.

Friends
std::ostream &	operator<< (std::ostream &os, const Matrix< T > &m)
	Stream output operator for formatted matrix display.

Detailed Description

template<typename T>
class Matrix< T >

A templated matrix class supporting various linear algebra operations.

This class provides a comprehensive implementation of matrix operations including basic arithmetic, advanced operations (determinant, inverse, eigenvalues), matrix decompositions (LU, QR), and iterative algorithms (Lanczos). Supports both real arithmetic types and complex number types.

Template Parameters

T	The element type. Can be arithmetic types (int, float, double) or complex number types (ComplexNumber).

Definition at line 22 of file matrix.h.

Constructor & Destructor Documentation

◆ Matrix() [1/3]

template<typename T >

Matrix< T >::Matrix ( )

inline

Default constructor creating an empty 0×0 matrix.

Definition at line 34 of file matrix.h.

◆ Matrix() [2/3]

template<typename T >

Matrix< T >::Matrix	(	int	r,
		int	c
	)

inline

Constructs a matrix with specified dimensions, initialized with default values.

Parameters

r	Number of rows
c	Number of columns

Definition at line 46 of file matrix.h.

◆ Matrix() [3/3]

template<typename T >

Matrix< T >::Matrix ( const std::vector< std::vector< T > > & mat )

Constructs a matrix from a 2D vector.

Parameters

mat	A 2D vector containing the matrix data

Exceptions

std::invalid_argument if rows have inconsistent lengths

Definition at line 113 of file matrix.cpp.

Member Function Documentation

◆ adjoint()

template<typename T >

Matrix< T > Matrix< T >::adjoint

Computes the adjoint (adjugate) matrix.

The adjoint is the transpose of the cofactor matrix.

Returns: Adjoint matrix

Exceptions

std::invalid_argument if matrix is not square

Definition at line 403 of file matrix.cpp.

◆ buildTridiagonal()

template<typename T >

Matrix< T > Matrix< T >::buildTridiagonal	(	const std::vector< T > &	alpha,
		const std::vector< T > &	beta,
		int	m
	)		const

Constructs a tridiagonal matrix from diagonal and off-diagonal elements.

Parameters

alpha	Diagonal elements
beta	Off-diagonal elements
m	Dimension of the tridiagonal matrix

Returns: m × m tridiagonal matrix

Definition at line 847 of file matrix.cpp.

◆ determinant()

template<typename T >

T Matrix< T >::determinant

Computes the determinant of the matrix.

The calculation follows these rules based on the matrix dimension \( n \times n \):

For \( n = 1 \):
\[ \det(A) = a_{1,1} \]
For \( n = 2 \):
\[ \det(A) = a_{0,0}a_{1,1} - a_{0,1}a_{1,0} \]
For \( n > 2 \), using LU Decomposition \( A = LU \):
\[ \det(A) = \det(L)\det(U) = \prod_{i=0}^{n-1} u_{i,i} \]
*(Note: \( \det(L) = 1 \) as it is a unit triangular matrix)*.

Returns: Determinant value of type \( T \).

Exceptions

std::invalid_argument If the matrix is not square ( \( rows \neq cols \)).

Definition at line 276 of file matrix.cpp.

◆ dot()

template<typename T >

T Matrix< T >::dot	(	const std::vector< T > &	a,
		const std::vector< T > &	b
	)		const

Computes the dot product of two vectors.

Parameters

a	First vector
b	Second vector

Returns: Dot product value

Exceptions

std::invalid_argument if vectors have different sizes

Definition at line 733 of file matrix.cpp.

◆ eigenvalues()

template<typename T >

std::vector< T > Matrix< T >::eigenvalues

Computes the eigenvalues of the matrix.

Returns: Vector containing eigenvalues

Exceptions

std::invalid_argument if matrix is not square

Definition at line 427 of file matrix.cpp.

◆ eigenvectors()

template<typename T >

Matrix< T > Matrix< T >::eigenvectors

Computes the eigenvectors of the matrix.

Returns: Matrix where each column is an eigenvector

Exceptions

std::invalid_argument if matrix is not square

Definition at line 865 of file matrix.cpp.

◆ frobeniusNorm()

template<typename T >

double Matrix< T >::frobeniusNorm

Computes the Frobenius norm of the matrix.

The Frobenius norm is the square root of the sum of squares of all elements.

Returns: Frobenius norm value

Definition at line 619 of file matrix.cpp.

◆ getCols()

template<typename T >

int Matrix< T >::getCols ( ) const

inline

Gets the number of columns in the matrix.

Returns: Number of columns

Definition at line 425 of file matrix.h.

◆ getMinor()

template<typename T >

Matrix< T > Matrix< T >::getMinor	(	int	row,
		int	col
	)		const

Gets the minor matrix by removing a specified row and column.

Used in computing determinants and adjoints via cofactor expansion.

Parameters

row	Row to remove (0-based)
col	Column to remove (0-based)

Returns: (rows-1) × (cols-1) minor matrix

Exceptions

std::out_of_range if row or column index is out of bounds

Definition at line 655 of file matrix.cpp.

◆ getRows()

template<typename T >

int Matrix< T >::getRows ( ) const

inline

Gets the number of rows in the matrix.

Returns: Number of rows

Definition at line 418 of file matrix.h.

◆ identity()

template<typename T >

Matrix< T > Matrix< T >::identity ( int n )

static

Creates an n × n identity matrix.

The identity matrix has ones on the main diagonal and zeros elsewhere.

Parameters

n	Dimension of the identity matrix

Returns: n × n identity matrix

Definition at line 706 of file matrix.cpp.

◆ inverse()

template<typename T >

Matrix< T > Matrix< T >::inverse

Computes the inverse of the matrix.

Returns: Inverse matrix

Exceptions

std::invalid_argument	if matrix is not square
std::runtime_error	if matrix is singular (determinant is zero)

Definition at line 303 of file matrix.cpp.

◆ isSymmetric()

template<typename T >

bool Matrix< T >::isSymmetric

Checks if the matrix is symmetric.

A matrix is symmetric if A = A^T.

Returns: true if matrix is symmetric, false otherwise

Exceptions

std::invalid_argument if matrix is not square

Definition at line 258 of file matrix.cpp.

◆ lanczos()

template<typename T >

void Matrix< T >::lanczos	(	int	m,
		std::vector< T > &	alpha,
		std::vector< T > &	beta,
		Matrix< T > &	Q
	)		const

Performs the Lanczos algorithm for tridiagonalization.

The Lanczos algorithm is an iterative method for computing eigenvalues and eigenvectors of large sparse symmetric matrices. It reduces the matrix to tridiagonal form.

Parameters

m	Number of Lanczos iterations
alpha	Output vector of diagonal elements (size m)
beta	Output vector of off-diagonal elements (size m-1)
Q	Output matrix of orthonormal Lanczos vectors (n × m)

Exceptions

std::invalid_argument if matrix is not square or m > matrix size

Definition at line 792 of file matrix.cpp.

◆ LUDecomposition()

template<typename T >

std::pair< Matrix< T >, Matrix< T > > Matrix< T >::LUDecomposition

Performs LU decomposition of the matrix.

Decomposes the matrix into lower triangular (L) and upper triangular (U) matrices such that A = LU.

Returns: Pair of matrices (L, U)

Exceptions

std::invalid_argument	if matrix is not square
std::runtime_error	if decomposition fails (singular matrix)

Definition at line 533 of file matrix.cpp.

◆ matVec()

template<typename T >

std::vector< T > Matrix< T >::matVec ( const std::vector< T > & v ) const

Performs matrix-vector multiplication.

Parameters

v	Vector to multiply with

Returns: Resulting vector after multiplication

Exceptions

std::invalid_argument if vector size doesn't match matrix columns

Definition at line 721 of file matrix.cpp.

◆ norm()

template<typename T >

double Matrix< T >::norm ( const std::vector< T > & v ) const

Computes the Euclidean (L2) norm of a vector.

Parameters

v	Vector to compute norm for

Returns: L2 norm value

Definition at line 743 of file matrix.cpp.

◆ normalize()

template<typename T >

void Matrix< T >::normalize ( std::vector< T > & v ) const

Normalizes a vector to unit length (in-place).

Parameters

v	Vector to normalize

Exceptions

std::runtime_error if vector has zero norm

Definition at line 749 of file matrix.cpp.

◆ ones()

template<typename T >

Matrix< T > Matrix< T >::ones	(	int	r,
		int	c
	)

static

Creates an r × c matrix filled with ones.

Parameters

r	Number of rows
c	Number of columns

Returns: r × c matrix of ones

Definition at line 909 of file matrix.cpp.

◆ operator()() [1/2]

template<typename T >

T & Matrix< T >::operator()	(	int	row,
		int	column
	)

Accesses matrix element at specified position (non-const).

Parameters

row	Row index (0-based)
column	Column index (0-based)

Returns: Reference to the element at (row, column)

Definition at line 134 of file matrix.cpp.

◆ operator()() [2/2]

template<typename T >

const T & Matrix< T >::operator()	(	int	row,
		int	column
	)		const

Accesses matrix element at specified position (const).

Parameters

row	Row index (0-based)
column	Column index (0-based)

Returns: Const reference to the element at (row, column)

Definition at line 146 of file matrix.cpp.

◆ operator*() [1/2]

template<typename T >

Matrix< T > Matrix< T >::operator* ( const Matrix< T > & other ) const

Matrix multiplication.

Parameters

other Matrix to multiply with

Returns: Result of matrix multiplication

Exceptions

std::invalid_argument if dimensions are incompatible (this.cols != other.rows)

Definition at line 190 of file matrix.cpp.

◆ operator*() [2/2]

template<typename T >

Matrix< T > Matrix< T >::operator* ( const T & scalar ) const

Scalar multiplication.

Parameters

scalar Scalar value to multiply all elements by

Returns: Result of scalar multiplication

Definition at line 211 of file matrix.cpp.

◆ operator+()

template<typename T >

Matrix< T > Matrix< T >::operator+ ( const Matrix< T > & other ) const

Matrix addition.

Parameters

other Matrix to add

Returns: Result of element-wise addition

Exceptions

std::invalid_argument if matrices have different dimensions

Definition at line 158 of file matrix.cpp.

◆ operator-()

template<typename T >

Matrix< T > Matrix< T >::operator- ( const Matrix< T > & other ) const

Matrix subtraction.

Parameters

other Matrix to subtract

Returns: Result of element-wise subtraction

Exceptions

std::invalid_argument if matrices have different dimensions

Definition at line 174 of file matrix.cpp.

◆ operator-=()

template<typename T >

Matrix< T > Matrix< T >::operator-= ( const Matrix< T > & other ) const

In-place subtraction.

Parameters

other Matrix to subtract

Returns: Result of element-wise subtraction

Exceptions

std::invalid_argument if matrices have different dimensions

◆ operator/=()

template<typename T >

Matrix< T > Matrix< T >::operator/= ( const Matrix< T > & other ) const

Element-wise division.

Parameters

other Matrix to divide by

Returns: Result of element-wise division

Exceptions

std::invalid_argument	if matrices have different dimensions
std::runtime_error	if division by zero occurs

◆ operator[]() [1/2]

template<typename T >

std::vector< T > & Matrix< T >::operator[] ( int i )

inline

Accesses a row of the matrix (non-const).

Parameters

i	Row index (0-based)

Returns: Reference to the row vector

Definition at line 85 of file matrix.h.

◆ operator[]() [2/2]

template<typename T >

const std::vector< T > & Matrix< T >::operator[] ( int i ) const

inline

Accesses a row of the matrix (const).

Parameters

i	Row index (0-based)

Returns: Const reference to the row vector

Definition at line 93 of file matrix.h.

◆ QRDecomposition()

template<typename T >

std::pair< Matrix< T >, Matrix< T > > Matrix< T >::QRDecomposition

Performs QR decomposition of the matrix using Gram-Schmidt orthogonalization.

Decomposes the matrix into an orthogonal matrix (Q) and an upper triangular matrix (R) such that A = QR.

Returns: Pair of matrices (Q, R)

Exceptions

std::invalid_argument if matrix has more columns than rows

Definition at line 563 of file matrix.cpp.

◆ rank()

template<typename T >

T Matrix< T >::rank

Computes the rank of the matrix.

The rank is the number of linearly independent rows or columns.

Returns: Rank of the matrix

Definition at line 223 of file matrix.cpp.

◆ spectralNorm()

template<typename T >

double Matrix< T >::spectralNorm

Computes the spectral norm (2-norm) of the matrix.

The spectral norm is the largest singular value, equal to the square root of the largest eigenvalue of A^T * A.

Returns: Spectral norm value

Definition at line 633 of file matrix.cpp.

◆ submatrix()

template<typename T >

Matrix< T > Matrix< T >::submatrix	(	int	row_start,
		int	row_end,
		int	col_start,
		int	col_end
	)		const

Extracts a submatrix from specified row and column ranges.

Extracts rows [row_start, row_end) and columns [col_start, col_end) using exclusive upper bounds.

Parameters

row_start	Starting row index (inclusive)
row_end	Ending row index (exclusive)
col_start	Starting column index (inclusive)
col_end	Ending column index (exclusive)

Returns: Extracted submatrix

Exceptions

std::out_of_range if indices are out of bounds

Definition at line 763 of file matrix.cpp.

◆ toString()

template<typename T >

std::string Matrix< T >::toString

Converts the matrix to a formatted string representation.

Returns: String representation of the matrix

Definition at line 684 of file matrix.cpp.

◆ transpose()

template<typename T >

Matrix< T > Matrix< T >::transpose

Computes the transpose of the matrix.

Returns: Transposed matrix with dimensions swapped

Definition at line 391 of file matrix.cpp.

◆ zeros()

template<typename T >

Matrix< T > Matrix< T >::zeros	(	int	r,
		int	c
	)

static

Creates an r × c matrix filled with zeros.

Parameters

r	Number of rows
c	Number of columns

Returns: r × c zero matrix

Definition at line 899 of file matrix.cpp.

Friends And Related Function Documentation

◆ operator<<

template<typename T >

std::ostream & operator<<	(	std::ostream &	os,
		const Matrix< T > &	m
	)

friend

Stream output operator for formatted matrix display.

Outputs the matrix with fixed-point notation, 6 decimal places, and field width of 12.

Parameters

os	Output stream
m	Matrix to output

Returns: Reference to the output stream

Definition at line 162 of file matrix.h.

The documentation for this class was generated from the following files:

source/include/matrix.h
source/matrix.cpp

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

Constructor & Destructor Documentation

◆ Matrix() [1/3]

◆ Matrix() [2/3]

◆ Matrix() [3/3]

Member Function Documentation

◆ adjoint()

◆ buildTridiagonal()

◆ determinant()

◆ dot()

◆ eigenvalues()

◆ eigenvectors()

◆ frobeniusNorm()

◆ getCols()

◆ getMinor()

◆ getRows()

◆ identity()

◆ inverse()

◆ isSymmetric()

◆ lanczos()

◆ LUDecomposition()

◆ matVec()

◆ norm()

◆ normalize()

◆ ones()

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ operator*() [1/2]

◆ operator*() [2/2]

◆ operator+()

◆ operator-()

◆ operator-=()

◆ operator/=()

◆ operator[]() [1/2]

◆ operator[]() [2/2]

◆ QRDecomposition()

◆ rank()

◆ spectralNorm()

◆ submatrix()

◆ toString()

◆ transpose()

◆ zeros()

Friends And Related Function Documentation

◆ operator<<