Define a matrix square root that preserves regularity

Question

Let $A:\mathbb{R}\to \mathbb{R}^{n\times m}$ and $B\in \mathbb{R}^{n\times k}$. Is it possible to define $C:\mathbb{R}\to \mathbb{R}^{n\times m}$ satisfying the following two properties:

1. for all $t\in \mathbb{R}$, $C_tC_t^\top=A_tA^\top_t+BB^\top$,
2. if $A$ is Lipschitz in $t$, i.e., $|A_t-A_s|\le L|t-s|$ for all $t,s$, then $C$ is also Lipschitz in $t$.

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The answer is affirmative for one-dimensional case, as one can simply define $C_t=\sqrt{A^2_t+B^2}$. Then one can easily verify that if $A$ is Lipschitz continuous, then $C$ is Lipschitz continuous regardless of $B=0$ or not.

The question is more challenging in a multidimensional setting, as $BB^\top$ has zero eigenvalue is not equivalent to $BB^\top =0$. It is not clear to me how to establish Item 2. I am happy to impose a boundedness condition on $A$.

Answer 1

You can use $$C_t=\sqrt{A_tA^\top_t+BB^\top}$$ as the positive square root of the positive definite matrix $A_tA^\top_t+BB^\top$.

You can find more discussions on related results searching for "\(BB^\top\) positive square root" on SearchOnMath, like this "Fractional power of self adjoint operators.".