Can I show that for any matrix A = ST, S & T Both symmetric and invertible matrices

Question

In my linear algebra class, my professor gave us an exercise, which is the following.

If A is an nxn matrix over the complex, one can show that A = ST for S and T both invertible and symmetric.

I guess I can use Jordan Block, but I don't know the exact step.

Can someone help me?

Answer 1

Every Jordan block can be factorized into two symmetric matrices, 3 by 3 for instance. $$ \pmatrix{ &&\lambda \\ & \lambda &\\ \lambda&& } \cdot \pmatrix{ &&1\\ &1 &1\\ 1&1& } $$

Now use the fact that conjugation of a symmetric matrix is still symmetric.

Moreover, if $A$ is nonsingular, then so are $S$ and $T$.