Welcome to MathOverflow

Question

1

is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ?

thanks a lot!

flag	asked Dec 20 at 11:57 David liu 19●1

1

what does the 'sum' notation mean? – suVRit Dec 20 at 11:59

1

and what does $A'$ mean? The transpose? The Hermitian transpose? – Dima Pasechnik Dec 20 at 12:15

1

I guess $A′$ means the transpose. Steven shows the eigenvectors are generally different. However, there is an interesting relation between the eigenvalues, under a mild assumption. See Corollary 2.2. of Lin & Wolkowicz, An eigenvalue majorization inequality for positive semidefinite block matrices, Linear Multilinear Algebra, 60 (2012), 1365-1368. – Betrand Dec 20 at 20:27

Steven Landsburg · Answer 1 · 2012-12-20 20:10:10Z UTC

By way of penance for my earlier "answer":

Take $A=\pmatrix{1&0\cr x&0\cr}$ and $B=\pmatrix{1&y\cr 0&0\cr}$.

Then the eigenvectors of $M=AA'+BB'$ and $N=A'A+B'B$ are in general different. As $x$ goes to 0, the eigenvectors of $M$ go off to zero and infinity while the eigenvectors of $N$ can be anything; as $y$ goes to 0, the eigenvectors of $N$ go off to zero and infinity while the eigenvectors of $M$ can be anything.

Steven Landsburg · Answer 2 · 2012-12-20 17:53:59Z UTC

Assuming that $sum(AA'+BB')$ means $AA'+BB'$ and that $A'$ means the transpose of $A$:

Let $x$ and $y$ be arbitrary complex numbers. Then the matrices $X=\pmatrix{0&1\cr 1&x\cr}$ and $Y=\pmatrix{0&1\cr1&y\cr}$ have arbitrary eigenvectors. But in general, the equaions $$\matrix{AA'+BB'=X&A'A+B'B==Y}$$ are solvable for $A,B$. So the answer to your question is no.

Edit: As Terry Tao points out in comments, this system of equations is clearly not solvable (just take traces). So this is not an answer to your question.

David liu · Answer 3 · 2012-12-21 04:53:32Z UTC

0

SORRY FOR come back later....

A is real matrix. sum(AA′+BB′) means AA′+BB′ and that A′ means the transpose of A.

thanks

edited Dec 21 at 4:53

answered Dec 21 at 4:44

David liu
1●1

is there any relationship between the eigenvector of sum(AA'+BB') and sum(A’A+B’B) ?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.