Effects of unitarian multiplication into the spectrum of a finite matrix. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:49:21Z http://mathoverflow.net/feeds/question/110321 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110321/effects-of-unitarian-multiplication-into-the-spectrum-of-a-finite-matrix Effects of unitarian multiplication into the spectrum of a finite matrix. Ricardo Marino 2012-10-22T11:55:42Z 2012-10-22T19:54:24Z <p>I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$? </p> <p>This matrix "square-root" has of course no unique solution, for if $P$ is a solution, $PU$ is also a solution if $U$ is unitary. If we consider the space $M_n(\mathbb{C})/U(n)$ we can seek an unique answer $P_*$, but I could not determine the shape of the space spanned by the eigenvalues of the equivalence class of $P_*$, apart for $n=2$ and some pretty horrible non-linear equations involving too many variables. Spectral theory is not my strong point, so I was wondering if anyone knew an answer to this: can we say what happens to the eigenvalues of $P$ if we multiply it by $U$ unitary?</p> http://mathoverflow.net/questions/110321/effects-of-unitarian-multiplication-into-the-spectrum-of-a-finite-matrix/110352#110352 Answer by Carlo Beenakker for Effects of unitarian multiplication into the spectrum of a finite matrix. Carlo Beenakker 2012-10-22T18:11:52Z 2012-10-22T19:54:24Z <p>By specifying $PP^{\dagger}=W$ you are prescribing the singular values $s_n$ ($n=1,2,\ldots N$) of the $N\times N$ matrix $P$. These are just the positive square roots of the eigenvalues of the Hermitian, nonnegative matrix $W$. So your question can be rephrased as, what is the relation between the eigenvalues $\lambda_n$ and the singular values $s_n$ of the matrix $P$. This is a classic problem studied by Horn (1954), who showed that the only relationships one can state in full generality are those obtained by Weyl (1949):</p> <p>$\prod_{n=1}^{k} |\lambda_n| \leq \prod_{n=1}^{k} s_n,$ for $k\lt N$, and $\prod_{n=1}^{N} |\lambda_n| = \prod_{n=1}^{N} s_n,$</p> <p>for the ordering $|\lambda_1|\gt|\lambda_2\gt\cdots\gt|\lambda_N|$ and $s_1\gt s_2\gt\cdots\gt s_N$. (The equality for $k=N$ follows trivially by equating the determinant of $PP^\dagger$ with the determinant of $W$.)</p> <p>More can be said for random matrices. As shown by Guionnet, Krishnapur, and Zeitouni (arXiv:0909.2214) in a probabilistic sense for “typical matrices”, the singular values almost determine the eigenvalues. In particular, the "single ring theorem" relates the eigenvalue density to the density of singular values.</p> <p>This <A HREF="http://rauhut.ins.uni-bonn.de/WSRMSlides/Rudelson.pdf" rel="nofollow">presentation</A> by Mark Rudelson gives an introduction.</p> <p>References:</p> <p>A. Horn, <em>On the eigenvalues of a matrix with prescribed singular values</em>, Proc. Amer. Math. Soc. <strong>5</strong>, 4–7, (1954).</p> <p>H. Weyl, <em>Inequalities between the two kinds of eigenvalues of a linear transformation</em>, Proc. Nat. Acad. Sci. USA <strong>35</strong>, 408–411, (1949).</p>