Measure of Subspace of Matrices with repeated Singular Values - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:23:30Z http://mathoverflow.net/feeds/question/76245 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76245/measure-of-subspace-of-matrices-with-repeated-singular-values Measure of Subspace of Matrices with repeated Singular Values Ashin 2011-09-23T23:45:14Z 2011-09-24T11:57:07Z <p>Hi All,</p> <p>Let us consider a P x Q real matrix (P >= Q). It can be thought of as an element of $\mathbb{R}^{PQ}$. We are considering Lebesgue measure over that space. My question is whether the subspace of matrices with repeated singular values are of measure 0 or not. </p> <p>Any suggestions would be welcome.</p> <p>Thanks Ashin</p> http://mathoverflow.net/questions/76245/measure-of-subspace-of-matrices-with-repeated-singular-values/76249#76249 Answer by Chris Godsil for Measure of Subspace of Matrices with repeated Singular Values Chris Godsil 2011-09-24T00:24:57Z 2011-09-24T00:24:57Z <p>The eigenvalues of <code>$$\widehat M = \begin{pmatrix}0&amp;M\\ M^T&amp;0\end{pmatrix}$$</code> are the squares of the singular values of $M$. A symmetric $n\times n$ matrix $N$ has a repeated eigenvalue if and only if the rank of $$N\otimes I - I\otimes N$$ is less than $n^2-n$. So the set of matrices $\widehat M$ with a repeated eigenvalue is a proper subvariety of the set of matrices $\widehat M$, hence this set will have measure zero.</p> http://mathoverflow.net/questions/76245/measure-of-subspace-of-matrices-with-repeated-singular-values/76260#76260 Answer by Ashin for Measure of Subspace of Matrices with repeated Singular Values Ashin 2011-09-24T03:33:41Z 2011-09-24T03:33:41Z <p>Hi Chris,</p> <p>Thanks a lot for your quick response. But I'm not that familiar with the results that you are using to arrive at the conclusion. Would you kindly be a bit more elaborate. Or maybe you can guide me to the resources where I could find the results that you are using here.</p> <p>Thanks a lot for your help. Ashin</p> http://mathoverflow.net/questions/76245/measure-of-subspace-of-matrices-with-repeated-singular-values/76263#76263 Answer by Denis Serre for Measure of Subspace of Matrices with repeated Singular Values Denis Serre 2011-09-24T07:59:25Z 2011-09-24T11:57:07Z <p>The set of matrices with a repeated eigenvalue is defined by an algebraic equation ${\rm disc}(M)=0$. This is the discriminant in the eigenvalues $$\prod_{i\lt j}(\lambda_j-\lambda_i)^2,$$ which is a polynomial in the entries of $M$. Because this polynomial is non-trivial, your set is a non-trivial algebraic variety. In particular, it has zero measure and is closed.</p>