Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?

share|improve this question
1  
If $D$ is a multiple of the identity matrix, easy. Otherwise, no idea. –  Will Jagy Mar 9 '13 at 3:09
    
Yeah, but I'd like to know the case when $D$ is only a general diagonal matrix. –  Hugo Mar 9 '13 at 3:12
    
we can look the problem in another way, we know the eigen values of $A$ and since $D$ is diagonal matrix, we also know the eigenvalues of $D$. Knowing eigen values of two matrix, how can we deduce the eigen value of $A+D$. [There is an ambiguity moving from eigen value to matrix] –  Learner Mar 9 '13 at 3:20
    
@Learner In that case, only when $A$ and $D$ can be diagonalized simultaneously can we add their eigenvalues together to compute that of $B$. –  Hugo Mar 9 '13 at 3:34
    
@Hugo Agreed. Im just repharsing the question to find an answer. –  Learner Mar 9 '13 at 3:37
add comment

1 Answer

up vote 3 down vote accepted

Modulo a similarity, it is like asking: if I know the eigenvalues of $A'$ and if $D'$ is a diagonalizable matrix, what can I say about the eigenvalues of $A'+D'$. Nothing, I'm afraid. The diagonalizable matrices are dense, so the eigenvalues of your matrix $A'+D'$ could be pretty much anything when $D'$ ranges over diagonalizable matrices.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.