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.
Best Answer
Surprisingly (at least to me) the answer is no when $n\ge 3$. This was proved by Yves Benoist and me after I mentioned the problem in a talk at MSRI and Yves came up with a great idea.
It is enough to show that there is no uniform bound when $n=3$. Here is an elementary argument that involves little computation. I don't pay attention to getting exact constants.
For small positive $\epsilon$ define \begin{equation} x_1 = e_1 \quad \quad x_2 = e_1 + \epsilon e_2 \quad \quad x_3 = e_1 + \epsilon e_3 \end{equation}` For certain distinct non zero real $\lambda_i$, which will depend on $\epsilon$, let $T\in M_n(\mathbb{R})$ be defined by $Tx_i = \lambda_i x_i$. The dual basis to $(x_i)$ is defined by \begin{equation} f_1=e_1 - {1\over\epsilon} (e_2 + e_3) \quad \quad f_2={1\over\epsilon} e_2 \quad \quad f_3={1\over\epsilon} e_3 \end{equation}
So the transpose and adjoint of $T$ is defined by $T^*f_i = \lambda_i f_i$. If $S$ implements a similarity between $T$ and $T^*$, it is more or less clear that $\|S\|\cdot \|S^{-1}\| \to \infty$ as $\epsilon \to 0$ uniformly over all permissible choices of $\lambda_i$. (For me the easy way to see this is to semi-normalize the $f_i$ as \begin{equation} \tilde{f}_1=\epsilon e_1 - (e_2 + e_3) \quad \quad \tilde{f}_2= e_2 \quad \quad \tilde{f}_3= e_3 \end{equation} The similarity $S$ must be given by $Sx_i = a_i \tilde{f}_i$ for some non zero $a_i$ since the $\lambda_i$ are distinct, and if $\|S\| \vee \|S^{-1}\| \le (1/3) C$, then all $|a_i|$ and all $1/|a_i|$ are less than $C$. But $\|x_2-x_3\|=\sqrt{2}\epsilon$ and $\|a_2 \tilde{f}_2 - a_3 \tilde{f}_3 \| =\sqrt{|a_2|^2+|a_3|^2} > \sqrt{2} /C$, which forces $\|S\| >1/(C\epsilon)$.)
Next a trivial but (it seems) important point. For fixed $\epsilon$, you can choose the $\lambda_i$ close enough to one so that $T$ is as close to the identity as you want. So we can choose $\lambda_i$ so that $\|T\|=1$ and $\|T^{-1}\|< 1+\epsilon$; denote such a $T$ by $T_\epsilon$.
Write $T_\epsilon = H_\epsilon K_\epsilon$ with $H_\epsilon$, $K_\epsilon$ (complex) Hermitian and $\|H_\epsilon\|=1$. We want to see that $\|K_\epsilon\| \to \infty$ as $\epsilon \to 0$. Notice that $H_\epsilon$ and $K_\epsilon$ are non singular since no $\lambda_i$ is zero. So we have $H_\epsilon^{-1}T_\epsilon H_\epsilon = K_\epsilon H_\epsilon = T_\epsilon^*$ and hence, by the first part of the proof, $\|H_\epsilon^{-1}\| \to \infty$ as $\epsilon \to 0$. But $H_\epsilon^{-1} = K_\epsilon T_\epsilon^{-1}$, so $$\|H_\epsilon^{-1}\| \le \|K_\epsilon\| \|T_\epsilon^{-1}\| \le (1+\epsilon) \|K_\epsilon\|. $$