The Eigenvalue Decomposition

All variations of eigenvalue decompositions (symmetric and non-symmetric, real and complex) are represented by the EigenvalueDecompositionT class. It has no constructors. Instead, it is created by calling the GetEigenvalueDecomposition method on the matrix. This method has two overloads. The first overload has no arguments: The second overload takes a Boolean argument that specifies whether the contents of the matrix may be destroyed during the calculation of the eigenvalue decomposition.

var A = Matrix.CreateSymmetric(4, new[] {
    4.16,-3.12, 0.56,-0.10,
    0, 5.03,-0.83, 1.18,
    0,0, 0.76, 0.34,
    0,0,0, 1.18 },
    MatrixTriangle.Upper, MatrixElementOrder.RowMajor);
var eig = A.GetEigenvalueDecomposition();

Dim A = Matrix.CreateSymmetric(4, New Double() {
    4.16, -3.12, 0.56, -0.1,
    0, 5.03, -0.83, 1.18,
    0, 0, 0.76, 0.34,
    0, 0, 0, 1.18},
    MatrixTriangle.Upper, MatrixElementOrder.RowMajor)
Dim eig = A.GetEigenvalueDecomposition()
'

let A = Matrix.CreateSymmetric(4, 
         [|
            4.16;-3.12; 0.56;-0.10;
            0.00; 5.03;-0.83; 1.18;
            0.00; 0.00; 0.76; 0.34;
            0.00; 0.00; 0.00; 1.18 
         |], MatrixTriangle.Upper, MatrixElementOrder.RowMajor)
let eig = A.GetEigenvalueDecomposition()

The Decompose method performs the actual decomposition. This method makes a copy of the matrix if necessary. It then calls the appropriate LAPACK routine to perform the actual decomposition. This method is called by other methods as needed. You will rarely need to call it explicitly.

Once the decomposition is computed, a number of operations can be performed. However, unlike other decompositions, eigenvalue decompositions generally don't give an advantage when solving equations or computing the inverse. For these methods, the EigenvalueDecompositionT class simply defers to corresponding method of the base matrix.

var b = Matrix.Create(4, 2, new double[] {
    8.70, -13.35, 1.89, -4.14,
    8.30, 2.13, 1.61, 5.00 }, MatrixElementOrder.ColumnMajor);
var x = eig.Solve(b);
var invA = eig.GetInverse();
var detA = eig.GetDeterminant();

Dim b = Matrix.Create(4, 2, New Double() {
    8.7, -13.35, 1.89, -4.14,
    8.3, 2.13, 1.61, 5.0}, MatrixElementOrder.ColumnMajor)
Dim x0 = eig.Solve(b)
Dim invA = eig.GetInverse()
Dim detA = eig.GetDeterminant()
'

let b = Matrix.Create(4, 2, 
         [|
            8.70; -13.35; 1.89; -4.14;
            8.30; 2.13; 1.61; 5.00 
         |], MatrixElementOrder.ColumnMajor)
let x = eig.Solve(b)
let invA = eig.GetInverse()
let detA = eig.GetDeterminant()

Because real matrices may have complex eigenvalues and eigenvectors, the eigenvalue decomposition of a real matrix deserves some special attention.

For a symmetric matrix, all eigenvalues and eigenvectors are always real. The Eigenvalues property returns a vector containing the eigenvalues. The Eigenvectors property returns a matrix that has the corresponding eigenvectors as its columns:

var L = eig.Eigenvalues;
var X = eig.Eigenvectors;

Dim L = eig.Eigenvalues
Dim X = eig.Eigenvectors
'

let L = eig.Eigenvalues
let X = eig.Eigenvectors

A non-symmetric real matrix may have complex eigenvalues, which occur in complex conjugate pairs. The corresponding eigenvectors are also complex and also come in complex conjugate pairs. The Eigenvalues and Eigenvectors return only the real eigenvalues and eigenvectors. To get all eigenvalues and eigenvectors, use the ComplexEigenvalues and ComplexEigenvectors properties.

The RawEigenvectors property returns a matrix whose columns contain all the information about the eigenvectors. For real eigenvalues, the corresponding column contains the corresponding eigenvector. For complex eigenvalues, which always occur in complex conjugate pairs, two adjoining columns contain the real and imaginary elements of the corresponding eigenvectors.

var A = Matrix.Create(4, 4, new double[] {
    0.35, 0.09, -0.44, 0.25,
    0.45, 0.07, -0.33, -0.32,
    -0.14, -0.54, -0.03, -0.13,
    -0.17, 0.35, 0.17, 0.11}, MatrixElementOrder.ColumnMajor);
var eig = A.GetEigenvalueDecomposition();

var L = eig.Eigenvalues;
var X = eig.Eigenvectors;
var complexL = eig.ComplexEigenvalues;
var complexX = eig.ComplexEigenvectors;
var rawX = eig.RawEigenvectors;

Dim A = Matrix.Create(4, 4, New Double() {
    0.35, 0.09, -0.44, 0.25,
    0.45, 0.07, -0.33, -0.32,
    -0.14, -0.54, -0.03, -0.13,
    -0.17, 0.35, 0.17, 0.11}, MatrixElementOrder.ColumnMajor)
Dim eig = A.GetEigenvalueDecomposition()

Dim L = eig.Eigenvalues
Dim X = eig.Eigenvectors
Dim complexL = eig.ComplexEigenvalues
Dim complexX = eig.ComplexEigenvectors
Dim rawX = eig.RawEigenvectors
'

let A = Matrix.Create(4, 4, 
         [|
            0.35; 0.09; -0.44; 0.25;
            0.45; 0.07; -0.33; -0.32;
            -0.14; -0.54; -0.03; -0.13;
            -0.17; 0.35; 0.17; 0.11
         |], MatrixElementOrder.ColumnMajor)
let eig = A.GetEigenvalueDecomposition()

let L = eig.Eigenvalues
let X = eig.Eigenvectors
let complexL = eig.ComplexEigenvalues
let complexX = eig.ComplexEigenvectors
let rawX = eig.RawEigenvectors

The eigenvalues and eigenvalues of a complex matrix are also complex. The Eigenvalues property returns a vector containing the eigenvalues. The Eigenvectors property returns a matrix that has the corresponding eigenvectors as its columns.

The eigenvalues of a Hermitian matrix are real. This simply means that the imaginary component of the eigenvalue vector is identically zero.

The ComplexEigenvalues, ComplexEigenvectors, and RawEigenvectors properties are meaningless for complex eigenvalue decompositions.