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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.

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.

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# What is this decomposition called?

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.

I was wondering about the (possibly non-unique) factorization where $X$ is $U D^\frac{1}{2}$ and columns of $U$ are 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.