A Question About Eigenvalue

[hrhgroup]

Dear friend

Suppose B is a diagonal matrix, C is another matrix (not necessarily diagonal) and A = B*C.
Now what is the relationship between the eigenvalues of A, and eigenvalues of B and C?
Is the bellow relation correct?
eigenvalues(A)=eigenvalues(B)*eigenvalues(C)

If its possible please refer me to the proof of it, too.
Thanks.

 

[subharpe]

This is not. In fact I don't think any such relaion can be drawn. Why don't you try with 2×2 matrices? I got the eigen value equations for :λA, λB, λC and it does not seem to have a relation as such.

 

[pangpangfish]

The relationship will not hold in general (unless C is also a diagonal matrix). Just pick a random numerical example. Let

B=[1 0; 0 2]
C=[1 2; 3 4]
A=[1 2; 6 8]

It is easy to verify that

eigen(A) != eigen(B)*eigen(C)


Hope that helps
PPF

Dear friend

Suppose B is a diagonal matrix, C is another matrix (not necessarily diagonal) and A = B*C.
Now what is the relationship between the eigenvalues of A, and eigenvalues of B and C?
Is the bellow relation correct?
eigenvalues(A)=eigenvalues(B)*eigenvalues(C)

If its possible please refer me to the proof of it, too.
Thanks.