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

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

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

Method explanation

Eigenvector-Eigenvalue identity

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

$${\left|v_{i,j}\right|}^2\cdot \prod _{k=1,k\ne i}^n\left({\lambda }_i-{\lambda }_k\right)=\prod _{k=1}^{n-1}\left({\lambda }_i-{\lambda }_k\left(a_j\right)\right)$$

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

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