eigenvalues of symmetric matrix sums

Question

Suppose $A$ and $B$ are symmatric matrices: $A,B \in S^n$. Let $Y=A+B$.

What is the relationship between eigenvalues of $Y$ and eigenvalues of $A$ and $B$?

Or,

Does any nonsingular matrix $P$ exist such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal matrices in the same time?

Maybe I'm in the wrong direction. What I need to show in process of my homework problem is: Suppose $A$ and $B$ are positive semidefinite matrices: $A,B \in S^n_+$. And $Tr(A+B)=1$.

Let $Y=A-B$. I'm trying to show that ${\parallel Y \parallel}_{2*} \leq 1$.

${\parallel Y \parallel}_{2*} = \sum\limits_{i = 1}^n {|{\lambda _i}(Y)|}$ is the summation of the absoluate values of the eigenvalues of $Y$.

$A\to P^TAP$ doesn't preserve eigenvalues. $A,B$ are simultaneously diagonalizable if they commute. — Qidi, Feb 14 '15 at 00:05
@user159517 I have to set in this way, because of the requirement of the problem. The problem is Problem 5.41 in Professor Boyd's textbook: convex optimization (https://web.stanford.edu/~boyd/.../bv_cvxbook.pdf). — zxzx179, Feb 14 '15 at 00:05
@Qidi Sorry, a typo:( Yeah, I know A,B could be simultaneously diagonalizable if they commute. But here I think this does not necessarily hold. — zxzx179, Feb 14 '15 at 00:09
Okay. I was confused because in the first line you say "Let $Y = A + B$" — user159517, Feb 14 '15 at 00:09
http://math.stackexchange.com/questions/478998/a-property-of-positive-definite-matrices check this — Brian Ding, Feb 14 '15 at 00:10

user159517 · Accepted Answer · 2015-02-14T01:21:59.720

Assuming you know $\|\cdot\|_{2*}$ is a matrix norm (which it turns out to be after a bit of researching, though I wouldnt know how to prove it right now), it can be shown as follows: First remark that for positive semidefinite matrices $P$ we have $\|P\|_{2*} = \mathrm{tr}(P)$. This holds because the eigenvalues of positive semidefinite matrices are nonnegative and because for symmetric matrices, the trace is equal to the sum of the eigenvalues. Then we have

$$ \|A-B\|_{2*} \leq \|A\|_{2*} + \|B\|_{2*} = \mathrm{tr}(A) + \mathrm{tr}(B) = \mathrm{tr}(A+B) = 1 $$

eigenvalues of symmetric matrix sums

1 Answers1