What is this decomposition called? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:26:50Z http://mathoverflow.net/feeds/question/41808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41808/what-is-this-decomposition-called What is this decomposition called? ohai 2010-10-11T16:34:04Z 2010-10-11T18:59:21Z <p>Let $M$ be a positive semi-definite matrix, symmetric with real entries. Then $M$ can be written as $X X^T$. One way is by a Cholesky decomposition (unique for positive definite but not necessarily for positive semi-definite $M$). Also note that for any $X X^T$ decomposition, $Y Y^T$ is also a decomposition where $Y = X R$ with $R$ orthogonal.</p> <p>I was wondering about the (possibly non-unique) factorization where $X$ is $U D^\frac{1}{2}$ and columns of $U$ are orthonormal eigenvectors of $M$ and $D$ is the diagonal matrix of sorted eigenvalues of $M$. Is this called something? I've been informally calling this $X$ the 'square root' of $M$ but I know that this is wrong and I would like to know if there is a correct word for it.</p>