Probably a well-know question, but I haven't solved it, so I'll ask.


I can show that every matrix in $M\_2(\mathbb{R})$ is the sum of two squares of matrices in $M\_2(\mathbb{R})$.
If $n>2$, I can also show that every matrix in $M\_n(\mathbb{R})$ is the sum of three squares of matrices in $M\_n(\mathbb{R})$.


So my question is : Is every matrix in $M\_n(\mathbb{R})$ is the sum of two squares of matrices in $M\_n(\mathbb{R})$ (n>2)?



Waring's problem for matrices
The answer is YES if $n$ is even. But if $n$ is odd, then the answer is NO since $-I$ is not a sum of two squares.
See 


Griffin and Krusemeyer, Matrices as sums of squares, *Linear and Multilinear Algebra* **5** (1977/78), no. 1, 33-44


for the proofs of these facts and generalizations.


