if A and B are symmetrical real matrices.

C=A+B

rc, ra, and rb are eigen values of C, A, and B.

I found that for the first several eigen values, they could be approximated as rc~=ra+rb. Is there any reason?

Expect your answers!

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- Jul 23rd 2009, 04:31 AMchenuoAn interesting problem on C=A+B
if A and B are symmetrical real matrices.

C=A+B

rc, ra, and rb are eigen values of C, A, and B.

I found that for the first several eigen values, they could be approximated as rc~=ra+rb. Is there any reason?

Expect your answers! - Jul 23rd 2009, 10:26 AMHallsofIvy
[b]If[/b ] A and B have the same eigenvectors, then the eigenvectors of C

**are**the sums of corresponding egenvectors of A and B. "Corresponding" here meaning "for the same eigenvector". If $\displaystyle Av= \lambda_A v$ and $\displaystyle Bv= \lambda_B v$, then (A+B)v= Av+ Bv= \lambda_A v+ \lambda_B v= (\lambda_A+ \lambda_B)v[/tex].

Otherwise, I don't know how you are going to choose which eigenvalues of A and B to add to get eigenvalues of B. - Jul 23rd 2009, 12:41 PMchenuoRE: HallsofIvy
There's no any relationship between A and B and we can not assume that A and B have the same eigenvectors. The eigenvalues we chose are the largest ones. The equation rc=ra+rb does not hold excactly! It's just approximation.