It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M_n(\mathbb R)$ (real entries) can be written as the product $HK$ of two Hermitian matrices (complex entries). Of course, the pair $(H,K)$ is far from being unique, because the real dimension of $\mathbb H_n\times\mathbb H_n$ is $2n^2$, much larger than $n^2=\dim M_n(\mathbb R)$. The question is whether this factorization can be done in a stable manner:

Does there exist a finite constant $c_n$ such that, for every $A\in M_n(\mathbb R)$, the pair $(H,K)\in\mathbb H_n\times\mathbb H_n$ can be chosen so that $A=HK$ and $\|H\|\cdot\|K\|\le c_n\|A\|$ ?

Of course the answer does not depend on the choice of the matrix norm. Only the constant does.

It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M_n(\mathbb R)$ (real entries) can be written as the product $HK$ of two Hermitian matrices (complex entries). Of course, the pair $(H,K)$ is far from being unique, because the real dimension of $\mathbb H_n\times\mathbb H_n$ is $2n^2$, much larger than $n^2=\dim M_n(\mathbb R)$. The question is whether this factorization can be done in a stable manner:

Does there exist a finite constant $c_n$ such that, for every $A\in M_n(\mathbb R)$, the pair $(H,K)\in\mathbb H_n\times\mathbb H_n$ can be chosen so that $A=HK$ and $\|H\|\cdot\|K\|\le c_n\|A\|$ ?

Of course the answer does not depend on the choice of the matrix norm. Only the constant does.

Edit. I must mention, to my shame, that at the beginning of Chapter 6 of my book on matrices (Springer-Verlag, GTM 216), I pretend that $\mathbb H_n\times\mathbb H_n$ equals $M_n(\mathbb C)$; without proof of course. Thanks to Jean Gallier, who pointed it out.