Eigenvector-eigenvalue identity Calculator

The online calculator calculates the eigenvalues of the matrix with the Denton, Parke, Tao, Zhang approach. The algorithm from Denton, Parke, Zhang uses the eigenvector-eigenvalue identity. The eigenvector-eigenvalue identity dosn't need the solution of a equation system. The algorithm uses the sub-matrices to calculate the magnitude of the eigenvectors.

Matrix

A= a11a12a1N a21a22a2N aN1aN2aNN

Dimension of the matrix N =

Number of digits =

Maximal number of QR iterations =

Enter the matrix elements for matrix A: a1,1, a1,2, ...

Input Matrix

Step-by-step calculation of the eingenvectors with the Denton, Parke, Tao, Zhang approach.

Calculating the eigenvalues of matrix A using QR decomposition

Calculating the eigenvalues of the sub-matrices using QR decomposition

Calculating the eigenvectors

Method explanation

Eigenvector-Eigenvalue identity

If A is an n×n Hermitian matrix with eigenvalues λ1(A),…,λn(A) and i,j=1,…,n, then the j-th component vi,j of a unit eigenvector vi associated to the eigenvalue λi(A) is related to the eigenvalues λ1(aj),…,λn−1(aj) of the minor aj of A formed by removing the j-th row and column by the formula

vi,j2 k = 1 , k i n λ i - λ k = k = 1 n - 1 λ i - λ k a j

Reference: Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra; Peter B. Denton, Stephen J. Parke, Terence Tao, Xining Zhang

Related sites

Here is a list of of further useful sites:

Index Matrix Determinant Eigenvalues QR decomposition Solver Adjugate matrix Solver Inverse Matrix