Finding Eigenvectors for Two Matrices using the Generalized Jacobi Method

[hoshangmustafa]

TL;DR Summary

If I have two matrices A and B, how can I find an eigenvector for the two matrices?

If I have two matrices A and B, how can I find an eigenvector for the two matrices?

 

[anuttarasammyak]

Eigenvectors belong to a matrix. Matrix A has its eigenvectors. Matrix B has its eigenvectors.

 

[hoshangmustafa]

Hi,
I meant simultaneous diagonalization of two matrices.

 

[anuttarasammyak]

I do not think simultaneously diagonalizable matters in procedure of getting eigenvectors of A and B for each.

As examples in QM, x coordinate operator X and y coordinate operator Y are simultaneously diagonarizable.
$$XY=YX$$
X has eigenvectors of {|x>}. Y has eigenvectors of {|y>}.

$$XX^2=X^2X$$
X^2 has denenerated eigenbectors of |x> and |-x> for eigenvalue x^2

 

[hoshangmustafa]

Hi,
for example

K=2,1;1,2
M=2,0;0,0
use the generalized Jacobi method to calculate the eigensystem problem
KΦ=λMΦ

 

[anuttarasammyak]

As $$KM \neq MK$$, I am afraid that we cannot simultaneously diagonalize them. I might be wrong due to scarce knowledge on Jacobi method.

 

[hoshangmustafa]

K=1,-1;-1,1
M=2,1;1,2

 

[anuttarasammyak]

Then KM=MK. M=K+2I where I is identity matrix. Diagonalization of K by product of unitary matrix $P, P^{-1}$ would also diagonalize M. Why don't you try to get it ?