SymmetricMatrixEigendecomposition Method

public RealEigendecomposition Eigendecomposition()

Public Function Eigendecomposition As RealEigendecomposition

public:
RealEigendecomposition^ Eigendecomposition()

member Eigendecomposition : unit -> RealEigendecomposition 

Return Value

For a generic vector v and matrix M, Mv = u will point in some direction with no particular relationship to v. The eigenvectors of a matrix M are vectors z that satisfy Mz = λz, i.e. multiplying an eigenvector by a matrix reproduces the same vector, up to a prortionality constant λ called the eigenvalue.

For v to be an eigenvector of M with eigenvalue λ, (M - λI)z = 0. But for a matrix to anihilate any non-zero vector, that matrix must have determinant, so det(M - λI)=0. For a matrix of order N, this is an equation for the roots of a polynomial of order N. Since an order-N polynomial always has exactly N roots, an order-N matrix always has exactly N eigenvalues.

An alternative way of expressing the same relationship is to say that the eigenvalues of a matrix are its diagonal elements when the matrix is expressed in a basis that diagonalizes it. That is, given Z such that Z^-1MZ = D, where D is diagonal, the columns of Z are the eigenvectors of M and the diagonal elements of D are the eigenvalues.

Note that the eigenvectors of a matrix are not entirely unique. Given an eigenvector z, any scaled vector αz is an eigenvector with the same eigenvalue, so eigenvectors are at most unique up to a rescaling. If an eigenvalue is degenerate, i.e. there are two or more linearly independent eigenvectors with the same eigenvalue, then any linear combination of the eigenvectors is also an eigenvector with that eigenvalue, and in fact any set of vectors that span the same subspace could be taken as the eigenvector set corresponding to that eigenvalue.

The eigenvectors of a symmetric matrix are always orthogonal and the eigenvalues are always real. The transformation matrix Z is thus orthogonal (Z^-1 = Z^T).

Finding the eigenvalues and eigenvectors of a symmetric matrix is an O(N³) operation.

If you require only the eigenvalues, not the eigenvectors, of the matrix, the Eigenvalues method will produce them faster than this method.

Reference