# Regularity of decomposition of matrix-valued function

Here is the problem. Suppose I have a positive definite matrix-valued function $A\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$. Then we know that there is a matrix-valued function $B\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$ such that $A=B^TB$, where $B^T$ is the transpose of $B$. I wonder if $A\in C^k$, what is the best possible regularity of the matrix $B$? More precisely, if $A\in C^0$ or $A\in L^\infty$, is it true that $B\in L^\infty$ as well?

If one tries to diagonalized by smooth similarity transformations, then it might not work; see

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

• The square root of $A$ is well defined, and has the same regularity as $A$. – abx Sep 29 '15 at 7:17
• @abx: It is not very clear to me, see mathoverflow.net/questions/60533/… somewhat surprisingly. – Changyu Guo Sep 29 '15 at 7:48
• The keyword here is positive. – abx Sep 29 '15 at 8:06
• @abx: Is there a simple proof of this fact without using diagonalization to diagonal matrix? – Changyu Guo Sep 29 '15 at 8:11
• See Square root of a matrix in Wikipedia. – abx Sep 29 '15 at 8:28

I recommend that you apply the Cholesky algorithm leading to the matrix $B$, see https://en.wikipedia.org/wiki/Cholesky_decomposition. It will follow that the smoothness class of $B$ will be the class obtained from the smoothness class of $A$ by applying several times the square root operation to positive functions (this operation clearly may reduce smoothness of functions). In the case of classes $C^0$ or $L^\infty$, they will be preserved.