In summary, the conversation discusses the process of finding eigenvectors for two matrices, A and B, and the concept of simultaneous diagonalization. The example of x and y coordinate operators in Quantum Mechanics is used to explain the concept further. The conversation also mentions the use of the generalized Jacobi method to calculate the eigensystem problem for two matrices, but it is noted that it may not be applicable if the matrices do not commute. The conversation ends by suggesting to use a unitary matrix to diagonalize the matrices.
#1
hoshangmustafa
4
0
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?
Eigenvectors belong to a matrix. Matrix A has its eigenvectors. Matrix B has its eigenvectors.
#3
hoshangmustafa
4
0
Hi,
I meant simultaneous diagonalization of two matrices.
#4
anuttarasammyak
Gold Member
2,501
1,325
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.
[tex]XY=YX[/tex]
X has eigenvectors of {|x>}. Y has eigenvectors of {|y>}.
[tex]XX^2=X^2X[/tex]
X^2 has denenerated eigenbectors of |x> and |-x> for eigenvalue x^2
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#5
hoshangmustafa
4
0
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
#6
anuttarasammyak
Gold Member
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1,325
As [tex]KM \neq MK[/tex], I am afraid that we cannot simultaneously diagonalize them. I might be wrong due to scarce knowledge on Jacobi method.
#7
hoshangmustafa
4
0
K=1,-1;-1,1
M=2,1;1,2
#8
anuttarasammyak
Gold Member
2,501
1,325
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 ?