Waring's problem for matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:34:13Z http://mathoverflow.net/feeds/question/20874 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20874/warings-problem-for-matrices Waring's problem for matrices Portland 2010-04-09T23:45:35Z 2010-04-10T00:27:17Z <p>Probably a well-know question, but I haven't solved it, so I'll ask.</p> <p>I can show that every matrix in \$M_2(\mathbb{R})\$ is the sum of two squares of matrices in \$M_2(\mathbb{R})\$. If \$n>2\$, I can also show that every matrix in \$M_n(\mathbb{R})\$ is the sum of three squares of matrices in \$M_n(\mathbb{R})\$.</p> <p>So my question is : Is every matrix in \$M_n(\mathbb{R})\$ is the sum of two squares of matrices in \$M_n(\mathbb{R})\$ (n>2)?</p> http://mathoverflow.net/questions/20874/warings-problem-for-matrices/20876#20876 Answer by Bjorn Poonen for Waring's problem for matrices Bjorn Poonen 2010-04-10T00:27:17Z 2010-04-10T00:27:17Z <p>The answer is YES if \$n\$ is even. But if \$n\$ is odd, then the answer is NO since \$-I\$ is not a sum of two squares. See </p> <p>Griffin and Krusemeyer, Matrices as sums of squares, <em>Linear and Multilinear Algebra</em> <strong>5</strong> (1977/78), no. 1, 33-44</p> <p>for the proofs of these facts and generalizations.</p>