Matrix2 Class Reference

This class implements \( 2^2 \) matrix. More...

#include <matrix.h>

Public Member Functions
	Matrix2 ()
	Empty.
constexpr	Matrix2 (const double &)
	Creates a matrix with the same diagonal value.
	Matrix2 (const Vector2 &)
	Create a diagonal matrix with diagonal terms set to the vector entries.
	Matrix2 (const Vector2 &, const Vector2 &)
	Creates a column vector matrix.
constexpr	Matrix2 (const double &, const double &, const double &, const double &)
	Creates a matrix with a set of double values.
	Matrix2 (const double &, const double &, const double &)
	Create a symmetric matrix.
constexpr double &	operator[] (int)
	Direct access to the array of the matrix.
constexpr double	operator[] (int) const
	Overloaded.
constexpr double &	operator() (int, int)
	Get element (i,j) of the matrix.
constexpr double	operator() (int, int) const
	Overloaded.
Vector2	C (int) const
	Get the i-th column vector of the matrix.
Matrix2	operator- () const
	Returns the opposite of a matrix -A.
Matrix2 &	operator+= (const Matrix2 &)
	Destructive addition operator.
Matrix2 &	operator-= (const Matrix2 &)
	Destructive subtraction operator.
Matrix2 &	operator*= (const Matrix2 &)
	Destructive multiplication.
Matrix2 &	operator*= (const double &)
	Destructive multiplication operator.
Matrix2 &	operator/= (double)
	Destructive division operator.
Matrix2	Scaled (const Matrix2 &) const
	Hadamart product, which is the element-wise product.
Vector2	operator* (const Vector2 &) const
	Right multiply by a vector.
QString	ToGLSL () const
	Return a string expression.
Matrix2	T () const
	Transpose the matrix.
double	Determinant () const
	Computes the determinant of the matrix.
Matrix2	Inverse () const
	Computes the inverse of a matrix \( A^{-1} \).
double	Trace () const
	Compute the trace (sum of diagonal terms) of a matrix.
double	FrobeniusNorm () const
	Compute the Frobenius norm of the matrix.
Quadric	Characteristic () const
	Compute the characteristic polynomial.
double	SpectralNorm () const
	Compute the spectral norm of the matrix.
double	InfinityNorm () const
	Compute the infinity norm of the matrix.
void	EigenSolveSymmetric (double[2], Vector2[2]) const
	Compute the eigen values of the matrix.
Vector2	Eigen () const
	Compute the eigen values.

Static Public Member Functions
static Matrix2	Rotation (const double &)
	Create a rotation matrix.
static Matrix2	Rotation (Xoshiro256 &=Xoshiro256::X512)
	Create a random rotation matrix.
static Matrix2	Tensor2 (const double &)
	Create an anisotropic tensor matrix.
static Matrix2	Tensor2 (const double &, const double &)
	Create a tensor matrix.

Static Public Attributes
static const Matrix2	Null
	Null matrix.
static const Matrix2	Identity
	Identity matrix.
static const Matrix2	RotationHalfPi
	Half-pi rotation.
static const Matrix2	RotationOneThirdPi
	One third pi rotation (60°).

Protected Attributes
double	r [4] = { 1.0,0.0,0.0,1.0 }
	The array storing the coefficients of the matrix.

Friends
Matrix2	operator+ (const Matrix2 &u, const Matrix2 &v)
	Overloaded.
Matrix2	operator- (const Matrix2 &u, const Matrix2 &v)
	Overloaded.
Matrix2	operator* (const Matrix2 &u, const Matrix2 &v)
	Multiplication.
Matrix2	operator* (const Matrix2 &u, const double &a)
	Right multiply by a double.
Matrix2	operator* (const double &a, const Matrix2 &u)
	Left multiply by a double.
Matrix2	operator/ (const Matrix2 &u, const double &a)
	Right divide by a double.
Matrix2	operator/ (const double &x, const Matrix2 &a)
	Computes the inverse matrix and scales it by a double.
std::ostream &	operator<< (std::ostream &s, const Matrix2 &matrix)
	Overloaded.

Detailed Description

This class implements \( 2^2 \) matrix.

Operators have been overloaded to behave as expected. Thus it is possible to write:

Matrix2 a = Matrix2(Vector2(1.0, 2.0));  // Scale
Matrix2 b = Matrix2::Rotation(Math::Pi / 4.0); // Rotation matrix
Vector2 u(2.0, -1.0); // Vector
Matrix2 c = a * b;  // Matrix product
Vector2 v = c * u;  // Matrix vector product

Constructor & Destructor Documentation

Matrix2() [1/5]

Matrix2::Matrix2 ( const double & a )

inlineexplicitconstexpr

Creates a matrix with the same diagonal value.

Parameters

a	The diagonal value.

Matrix2() [2/5]

Matrix2::Matrix2 ( const Vector2 & a )

inlineexplicit

Create a diagonal matrix with diagonal terms set to the vector entries.

Note that this is a scaling matrix.

Parameters

a	Vector of diagonal values

Matrix2() [3/5]

Matrix2::Matrix2 ( const Vector2 & a, const Vector2 & b )

inlineexplicit

Creates a column vector matrix.

Parameters

a,b	Column vectors.

Matrix2() [4/5]

Matrix2::Matrix2 ( const double & c0r0, const double & c0r1, const double & c1r0, const double & c1r1 )

inlineexplicitconstexpr

Creates a matrix with a set of double values.

The coefficients are given column by column: the parameter cJrI is the coefficient of the column J and of the row I, i.e., the element (I,J) of the matrix, which is stored at the index I+2J. The matrix whose rows are (a,b) and (c,d) is therefore Matrix2(a, c, b, d).

Parameters

c0r0,c0r1	Coefficients of the first column.
c1r0,c1r1	Coefficients of the second column.

Matrix2() [5/5]

Matrix2::Matrix2 ( const double & a, const double & b, const double & c )

inlineexplicit

Create a symmetric matrix.

Parameters

a,b	Diagonal terms.
c	Last term.

Member Function Documentation

C()

Vector2 Matrix2::C ( int i ) const

inline

Get the i-th column vector of the matrix.

Parameters

i

Index.

EigenSolveSymmetric()

void Matrix2::EigenSolveSymmetric	(	double	a[2],
		Vector2	e[2] ) const

Compute the eigen values of the matrix.

Return eigen vectors as well.

Parameters

a	An array of two eigen values.
e	The array containing the two eigen vectors.

Inverse()

Matrix2 Matrix2::Inverse

(

)

const

Computes the inverse of a matrix \( A^{-1} \).

This function returns the null matrix if A cannot be inversed.

The threshold value involved in the singular matrix detection is set to \( 10^{-18} \).

operator()()

double & Matrix2::operator() ( int i, int j )

inlineconstexpr

Get element (i,j) of the matrix.

Parameters

i	Row.
j	Column.

operator+=()

Matrix2 & Matrix2::operator+=

(

const Matrix2 &

u

)

Destructive addition operator.

Parameters

u

Matrix.

operator-=()

Matrix2 & Matrix2::operator-=

(

const Matrix2 &

u

)

Destructive subtraction operator.

Parameters

u

Matrix.

Rotation() [1/2]

Matrix2 Matrix2::Rotation ( const double & a )

static

Create a rotation matrix.

Parameters

a	Angle in radian.

Rotation() [2/2]

Matrix2 Matrix2::Rotation ( Xoshiro256 & ra = Xoshiro256::X512 )

static

Create a random rotation matrix.

This is a simple function which is equivalent to the following inlined code:

Xoshiro256 r;
Matrix2 m=Matrix2::Rotation(2.0*Math::Pi*r.Uniform())

Parameters

ra	Random number generator.

Scaled()

Matrix2 Matrix2::Scaled

(

const Matrix2 &

u

)

const

Hadamart product, which is the element-wise product.

Parameters

u

Matrix.

T()

Matrix2 Matrix2::T ( ) const

inline

Transpose the matrix.

Note that if the matrix is a rotation matrix, then the transpose is equal to its inverse.

See also

Matrix::T(), Matrix4::T()

Tensor2() [1/2]

Matrix2 Matrix2::Tensor2 ( const double & angle )

static

Create an anisotropic tensor matrix.

Parameters

angle

The angle of the tensor.

Tensor2() [2/2]

Matrix2 Matrix2::Tensor2 ( const double & c, const double & s )

static

Create a tensor matrix.

Parameters

c,s	Cosine and sine of twice the tensor angle, i.e. \( c=\cos(2\theta) \), \( s=\sin(2\theta) \). Use Tensor2(double) to pass the angle \( \theta \) directly.

ToGLSL()

QString Matrix2::ToGLSL

(

)

const

Return a string expression.

Yields a QString of the form 'mat2(m(0, 0), ..., m(1, 1))' that write the matrix in GLSL.

Author

Hubert-Brierre Pierre

Friends And Related Symbol Documentation

operator*

Matrix2 operator* ( const Matrix2 & u, const Matrix2 & v )

friend

Multiplication.

Parameters

u,v	Input matrices.

operator/

Matrix2 operator/ ( const double & x, const Matrix2 & a )

friend

Computes the inverse matrix and scales it by a double.

Parameters

x	Real.
a	Matrix.

operator<<

std::ostream & operator<< ( std::ostream & s, const Matrix2 & matrix )

friend

Overloaded.

Parameters

s	Stream.
matrix	The matrix.