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So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\ldots\leq \alpha_n$, and the eigenvalues of $X$ as $\beta_1\leq\ldots\leq \beta_n$. Is is necessarily true that $\alpha_i\geq\beta_i$ for all $i$?

I might be able to solve this myself (although with time I am less sure), but it should be be much easier for whoever already knows the answer. A quick reference would do, thanks.

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Where does this question come from? –  Igor Rivin Sep 18 '12 at 16:34

3 Answers 3

up vote 9 down vote accepted

This is true and well known. By the minimax principle, $\alpha_k$ is the minimum over all $k$-dimensional subspaces of the norm of the quadratic form $v\mapsto(v,Yv)$ restricted to the subspace. And similarly for $\beta_k$ and $(v,Xv)$. Since $(v,Yv)\ge(v,Xv)$ for every vector $v$, the same inequality holds for the norms of restrictions to subspaces and hence for eigenvalues.

I think any textbook covering the minimax principle has this corollary.

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Yes, you are right, it's quite simple. Too bad I missed it.. While thinking of my question above I was sidetracked to think of a stronger related question: Suppose E is a ellipsoid contained inside a bigger ellipsoid F. Can E always be moved inside F until all their axes are parallel? A positive answer also gives a positive answer to my original question, but I don't see how to obtain an implication in the other direction. Do you happen to know the solution to the latter question as well? –  puzne Sep 18 '12 at 19:29

At this point I find it worth mentioning the following facts. Let $X$ and $Y$ be arbitrary square complex matrices. Let $|X|=(X^*X)^{1/2}$ be the matrix absolute value. Then, we may have the following inequalities (in decreasing order of strength, i.e., a higher one in the list implies a lower one):

  1. $|X| \succeq |Y|$
  2. $\sigma_i(X) \ge \sigma_i(Y)$, where $\sigma_i$ is the $i$th singular value
  3. $\prod_{j=1}^k \sigma_j^\downarrow(X) \ge \prod_{j=1}^k \sigma_j^\downarrow(Y)$ for $k=1,2,\ldots,n$;
  4. $\sum_{j=1}^k \sigma_j^\downarrow(X) \ge \sum_{j=1}^k \sigma_j^\downarrow(Y)$ for $k=1,2,\ldots,n$;
  5. $\| X\| \ge \|Y\|$, where $\|\cdot\|$ is any unitarily invariant norm.

Actually, in the list above, $4\Leftrightarrow 5$. The original question is a special case of $1 \implies 2$, and the textbook proof is as S. Ivanov has nicely explained above.

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Thanks - do you have an easy reference for the above facts? –  puzne Sep 19 '12 at 16:43
    
Hi Puzne, these results are developed in several books; I find the book "Matrix Analysis" by R. Bhatia, particularly readable. Notice that 1,2 follows from Sergei's answer (and the fact that $\sigma(|X|)=\sigma(X)$ ); then 2 obviously implies 3. The fact that $3$ implies $4$ is a short exercise, while $4$ implies $5$ is an important theorem in matrix analysis; so basically, you'll need to refer to some book only for the last result. –  Suvrit Sep 19 '12 at 18:22

No, but $$ \sum_{j=1}^{k} \alpha_j \geq \sum_{j=1}^{k} \beta_j $$ for all $k$ as EASILY follows from the mini max principle.

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