is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:00:57Z http://mathoverflow.net/feeds/question/116856 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116856/is-there-any-relationship-between-the-eigenvector-of-sumaabb-and-sumaabb is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? David liu 2012-12-20T11:57:50Z 2012-12-21T04:53:32Z <p>is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ?</p> <p>thanks a lot!</p> http://mathoverflow.net/questions/116856/is-there-any-relationship-between-the-eigenvector-of-sumaabb-and-sumaabb/116873#116873 Answer by Steven Landsburg for is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? Steven Landsburg 2012-12-20T15:37:50Z 2012-12-20T17:53:59Z <p>Assuming that \$sum(AA'+BB')\$ means \$AA'+BB'\$ and that \$A'\$ means the transpose of \$A\$:</p> <p>Let \$x\$ and \$y\$ be arbitrary complex numbers. Then the matrices \$X=\pmatrix{0&amp;1\cr 1&amp;x\cr}\$ and \$Y=\pmatrix{0&amp;1\cr1&amp;y\cr}\$ have arbitrary eigenvectors. But in general, the equaions \$\$\matrix{AA'+BB'=X&amp;A'A+B'B==Y}\$\$ are solvable for \$A,B\$. So the answer to your question is no.</p> <p><b>Edit:</b> As Terry Tao points out in comments, this system of equations is clearly <b>not</b> solvable (just take traces). So this is not an answer to your question.</p> http://mathoverflow.net/questions/116856/is-there-any-relationship-between-the-eigenvector-of-sumaabb-and-sumaabb/116892#116892 Answer by Steven Landsburg for is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? Steven Landsburg 2012-12-20T18:59:11Z 2012-12-20T20:10:10Z <p>By way of penance for my earlier "answer":</p> <p>Take \$A=\pmatrix{1&amp;0\cr x&amp;0\cr}\$ and \$B=\pmatrix{1&amp;y\cr 0&amp;0\cr}\$.</p> <p>Then the eigenvectors of \$M=AA'+BB'\$ and \$N=A'A+B'B\$ are in general different. As \$x\$ goes to 0, the eigenvectors of \$M\$ go off to zero and infinity while the eigenvectors of \$N\$ can be anything; as \$y\$ goes to 0, the eigenvectors of \$N\$ go off to zero and infinity while the eigenvectors of \$M\$ can be anything.</p> http://mathoverflow.net/questions/116856/is-there-any-relationship-between-the-eigenvector-of-sumaabb-and-sumaabb/116951#116951 Answer by David liu for is there any relationship between the eigenvector of sum(AA'+BB') and sum(A'A+B'B) ? David liu 2012-12-21T04:44:49Z 2012-12-21T04:53:32Z <p>SORRY FOR come back later....</p> <p>A is real matrix. sum(AA′+BB′) means AA′+BB′ and that A′ means the transpose of A.</p> <p>thanks</p>