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

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?

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?

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Changyu Guo
Changyu Guo
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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?

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?

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?

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Changyu Guo
Changyu Guo
  • 1.9k
  • 11
  • 17

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?