Principal Minors of Matrix Product - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:17:34Z http://mathoverflow.net/feeds/question/38434 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38434/principal-minors-of-matrix-product Principal Minors of Matrix Product Alex Lupsasca 2010-09-12T01:21:08Z 2010-09-12T02:36:35Z <p>Suppose $A$ is a positive definite matrix and $B$ is a non-symmetric matrix with all positive principal minors. </p> <p>Is their product $AB$ a matrix with all positive principal minors?</p> <p>I believe the answer is yes, and I have been trying to find a proof but got stuck along the way. The wiki page for minor gives a corollary to the Cauchy-Binet formula which I think may be of use: <a href="http://en.wikipedia.org/wiki/Minor_%28linear_algebra%29#Applications" rel="nofollow">http://en.wikipedia.org/wiki/Minor_%28linear_algebra%29#Applications</a></p> <p>Thank you, Alex</p> http://mathoverflow.net/questions/38434/principal-minors-of-matrix-product/38436#38436 Answer by Victor Protsak for Principal Minors of Matrix Product Victor Protsak 2010-09-12T02:36:35Z 2010-09-12T02:36:35Z <p>This isn't true even if $A$ and $B$ are both symmetric and positive definite. For example, let <code>$$A=\begin{pmatrix} 1 &amp; 2\\2 &amp; 5\end{pmatrix}, \quad B=\begin{pmatrix} 1 &amp; -2\\-2 &amp; 5\end{pmatrix},\quad\text{then}\quad AB=\begin{pmatrix} -3 &amp; 8\\-8 &amp; 21\end{pmatrix}.$$</code></p>