Convex Combination of 2 hermitian matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:34:33Z http://mathoverflow.net/feeds/question/109682 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109682/convex-combination-of-2-hermitian-matrices Convex Combination of 2 hermitian matrices dineshdileep 2012-10-15T05:04:37Z 2012-10-15T15:13:31Z <p>Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices are guaranteed to have at least one zero eigen value), does there exist any positive number $t$ such that equation \begin{align} (tA_1+(1−t)A_2)x=0 \end{align} has a non-zero vector $x$ as a solution.</p> <p>( This is my first question in mathoverflow. I am not a mathematician, but from an engineering back ground. In my application, this kind of problem arises. My level of mathematical maturity is not enough to solve it. I hope some one here can.)</p> http://mathoverflow.net/questions/109682/convex-combination-of-2-hermitian-matrices/109683#109683 Answer by Robert Israel for Convex Combination of 2 hermitian matrices Robert Israel 2012-10-15T05:09:23Z 2012-10-15T05:09:23Z <p>Trivially no: consider the case $A_1 = A_2$. </p> http://mathoverflow.net/questions/109682/convex-combination-of-2-hermitian-matrices/109704#109704 Answer by Dirk for Convex Combination of 2 hermitian matrices Dirk 2012-10-15T10:20:37Z 2012-10-15T15:07:11Z <p>I suspect that you mean <code>$0 &lt; t &lt; 1$</code> in your question.</p> <p>Then the answer in still no under the added condition that both matrices are not of full rank. Consider $$A_1 = \begin{pmatrix}1 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; -1\end{pmatrix}\quad A_2 = \begin{pmatrix}0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0\\ 0 &amp; 0 &amp; -1\end{pmatrix}.$$</p> http://mathoverflow.net/questions/109682/convex-combination-of-2-hermitian-matrices/109724#109724 Answer by Peter Michor for Convex Combination of 2 hermitian matrices Peter Michor 2012-10-15T15:13:31Z 2012-10-15T15:13:31Z <p>Consider the characteristic polynomial $$\lambda^N -a_1(t)\lambda^{N-1} + \dots +(-1)^{N} a_N(t) = 0$$ of your Hermitian matrix $tA_1 +(1-t)A_2$ which has all roots real and whose coefficients are real analytic as functions of $t$. By a theorem of Rellich, the roots can be arranged as real analytic functions of $t$ also. Moreover, the eigenvectors of $tA_1 + (1-t)A_2$ can also be arranged real analytic in $t$. See [Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998)] [(pdf)]<a href="http://www.mat.univie.ac.at/~michor/roots.pdf" rel="nofollow">1</a>. (2.5 is wrong in this paper.)</p> <p>By your assumptions you have at both $t=0$ and $t=1$: At least one positive root, one negative root, and one root zero.<br> From this follows nothing, and it is easy to come up with simple examples with any outcome.</p> <p>But if you can ascertain that a positive eigenvector for $A_1$ turns into a negative one for $A_2$, then the corresponding eigenvalue must go through 0 in between. </p> <p>Maybe this can help you playing with your assumptions.</p>