Algebraic Operations

Hereinafter the following convention is used:

– small Greek letters are used to denote scalars;
– small Latin letters are used to denote vectors;
– capital Latin letters are used to denote matrices.

This topic contains the following sections:

Unary operations
Addition
Subtraction
Multiplication
Division
Overloaded operators
See Also

Unary operations

This section describes methods implementing unary operations, i.e. operations performed with one matrix.

Operation	Description	Performance
plus	Creates a copy of the original matrix. Actually, PlusInPlace() method does nothing. In place: PlusInPlace Returning result: Plus Out of place: Plus(Matrix)
minus	Negates this matrix. If the matrix is , then the result of this operation will be In place: MinusInPlace Returning result: Minus Out of place: Minus(Matrix)

Operation

Description

Performance

plus

Creates a copy of the original matrix. Actually, PlusInPlace() method does nothing.

minus

Negates this matrix. If the matrix is MMatrix ,

then the result of this operation will be MMinus

Addition

Addition operation allows adding a matrix or a scalar to the given matrix.

Operation	Description	Performance
matrix-scalar addition	Calculates the sum of a scalar and a matrix (or a matrix and a scalar – as you know the addition is commutative operation). This operation is defined as: given: then: In place: AddInPlace(Double) Returning result: Add(Double) Out of place: Add(Double, Matrix)
matrix-matrix addition	Adds the given matrix to this matrix. This operation is defined as: given: then: In place: AddInPlace(Matrix) Returning result: Add(Matrix) Out of place: Add(Matrix, Matrix)

Operation

Description

Performance

matrix-scalar addition

Calculates the sum of a scalar and a matrix (or a matrix and a scalar – as you know the addition is commutative operation). This operation is defined as:

given:

then:

In place:

method AddInPlace(Double)

Returning result:

method Add(Double)

Out of place:

method Add(Double, Matrix)

matrix-matrix addition

Adds the given matrix to this matrix. This operation is defined as:

given:

then:

In place:

method AddInPlace(Matrix)

Returning result:

method Add(Matrix)

Out of place:

method Add(Matrix, Matrix)

Subtraction

Subtraction allows calculating the difference between a matrix and a scalar or between two matrices.

Operation	Description	Performance
matrix-scalar subtraction	The difference of a matrix and a scalar: In place: SubtractInPlace(Double) Returning result: Subtract(Double) Out of place: Subtract(Double, Matrix) The difference of a scalar and a matrix: In place: LeftSubtractInPlace(Double) Returning result: LeftSubtract(Double) Out of place: LeftSubtract(Double, Matrix)
matrix-matrix subtraction	The difference of the given matrix and another matrix: then: In place: SubtractInPlace(Matrix) Returning result: Subtract(Matrix) Out of place: Subtract(Matrix, Matrix) The opposite operation: In place: LeftSubtractInPlace(Matrix) Returning result: LeftSubtract(Matrix) Out of place: LeftSubtract(Matrix, Matrix)

Operation

Description

Performance

matrix-scalar subtraction

The difference of a matrix and a scalar:

In place:

method SubtractInPlace(Double)

Returning result:

method Subtract(Double)

Out of place:

method Subtract(Double, Matrix)

The difference of a scalar and a matrix:

In place:

method LeftSubtractInPlace(Double)

Returning result:

method LeftSubtract(Double)

Out of place:

method LeftSubtract(Double, Matrix)

matrix-matrix subtraction

The difference of the given matrix and another matrix:

then:

In place:

method SubtractInPlace(Matrix)

Returning result:

method Subtract(Matrix)

Out of place:

method Subtract(Matrix, Matrix)

The opposite operation:

In place:

method LeftSubtractInPlace(Matrix)

Returning result:

method LeftSubtract(Matrix)

Out of place:

method LeftSubtract(Matrix, Matrix)

Multiplication

Multiplication of a matrix by a scalar, a vector or a matrix.

Operation	Description	Performance
matrix-scalar multiplication	Multiplies this matrix by the given scalar: In place: MultiplyInPlace(Double) Returning result: Multiply(Double) Out of place: Multiply(Double, Matrix)
matrix-vector multiplication	Multiplies this matrix by the column vector from the right. Consider the given vector is: The result will be: Returning result: Multiply(Vector) Out of place: Multiply(Vector, Vector) Multiplies this matrix by the row vector from the left. Consider the given vector is: Returning result: LeftMultiply(Vector) Out of place: LeftMultiply(Vector, Vector)
matrix-matrix multiplication	Multiplies this matrix by the given matrix from the right. Consider the given matrix is: The result will be: Returning result: Multiply(Matrix) Out of place: Multiply(Matrix, Matrix) Multiplies this matrix by the given matrix from the left. Consider the given matrix is: The result will be: Returning result: LeftMultiply(Matrix) Out of place: LeftMultiply(Matrix, Matrix)

Operation

Description

Performance

matrix-scalar multiplication

Multiplies this matrix by the given scalar:

In place:

method MultiplyInPlace(Double)

Returning result:

method Multiply(Double)

Out of place:

method Multiply(Double, Matrix)

matrix-vector multiplication

Multiplies this matrix by the column vector from the right.

Consider the given vector is:

The result will be:

Returning result:

method Multiply(Vector)

Out of place:

method Multiply(Vector, Vector)

Multiplies this matrix by the row vector from the left.

Consider the given vector is:

Returning result:

method LeftMultiply(Vector)

Out of place:

method LeftMultiply(Vector, Vector)

matrix-matrix multiplication

Multiplies this matrix by the given matrix from the right.

Consider the given matrix is:

The result will be:

Returning result:

method Multiply(Matrix)

Out of place:

method Multiply(Matrix, Matrix)

Multiplies this matrix by the given matrix from the left.

Consider the given matrix is:

The result will be:

Returning result:

method LeftMultiply(Matrix)

Out of place:

method LeftMultiply(Matrix, Matrix)

Division

Division of a matrix by a scalar.

Operation	Description	Performance
matrix-scalar division	Divides each element of the matrix by the given scalar: In place: DivideInPlace(Double) Returning result: Divide(Double) Out of place: Divide(Double, Matrix)

Operation

Description

Performance

matrix-scalar division

Divides each element of the matrix by the given scalar:

In place:

method DivideInPlace(Double)

Returning result:

method Divide(Double)

Out of place:

method Divide(Double, Matrix)

Overloaded operators

The described algebraic operations methods can be replaced by mathematical signs of this operations: +, -, *, / like in example below:

Copy

1Matrix A = new Matrix(4);
2Matrix B = new Matrix(4);
3
4// Sum of matrices.
5Matrix C = A + B;
6
7// Sum of a matrix and a scalar.
8Matrix D = A + 2.0;

Other Resources

Matrix Class

Pointwise Operations

Copying And Conversion

Transposition, Normalization, Inversion

Rank, Determinan, Condition Number, Trace

Matrix Norms

Triangles And Trapezoids

Permutations

Floating Point Arithmetic

Stacking Matrices

Extensions

Input And Output