# 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

$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1N}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2N}\\ & ⋮\\ {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right)$

↹#.000
Matrix dimension N =
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.

## 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

$vi,j2⋅ ∏ k = 1 , k ≠ i n λ i - λ k = ∏ k = 1 n - 1 λ i - λ k a j$

## Related sites

Here is a list of of further useful sites:

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