SchurDecomposition<T> Class

public abstract class SchurDecomposition<T> : Decomposition<T>

Public MustInherit Class SchurDecomposition(Of T)
	Inherits Decomposition(Of T)

generic<typename T>
public ref class SchurDecomposition abstract : public Decomposition<T>

[<AbstractClassAttribute>]
type SchurDecomposition<'T> = 
    class
        inherit Decomposition<'T>
    end

Inheritance

Object  →  LinearOperator<T>  →  Decomposition<T>  →  SchurDecomposition<T>

Derived

Numerics.NET.LinearAlgebra.ComplexDenseSchurDecomposition<T>

Type Parameters

T

The Schur decomposition of a real (complex) square matrix A rewrites the matrix in the form A= QTQ^*, where Q is an orthogonal (unitary) matrix and T, the Schur form, is an upper triangular matrix. The diagonal elements of T are the eigenvalues of A.

For a real matrix, the eigenvalues are either real or come in complex conjugate pairs. When there are complex eigenvalues, the matrix T is modified to be quasi-triangular, where 2x2 blocks on the diagonal indicate complex conjugate pairs of eigenvalues. This is sometimes referred to as the real Schur form.

The Schur form of a complex matrix is always upper triangular.

The Schur form of a real symmetric or complex Hermitian matrix is a diagonal matrix. Therefore, the Schur decomposition of such a matrix is the same as the eigenvalue decomposition.

SchurDecomposition<T> inherits from LinearOperator<T>. However, unlike most other decompositions, the Schur decomposition does not offer a speed advantage when performing most calculations. For example, solving an equation using the eigenvalue decomposition itself requires the solution of a system of equations. For this reason, the Solve(Vector<T>), GetInverse(Boolean) and EstimateConditionNumber() and similar methods simply call the corresponding method on the base matrix.

Reference