Spectra of two different symmetrizations of the same matrix $A$

Question

Let $A$ be an $n\times n$ real matrix. There are two easy ways to tranform it into a symmetric matrix:

1. take $B:=\frac{1}{2}(A+A^T)$
2. take $C:=\sqrt{AA^T}$

$B,C$ are both symmetric, hence they both have $n$ real eigenvalues. $C$ is non-negative definite and its eigenvalues are called the singular values of $A$.

In general is there a relation between the eigenvalues of $B$ and those of $C$ ? Do they have the same spectral radius?

For example if the matrix $A$ is symmetric itself, then one has simply $B=A$, $C^2=A^2$ hence the eigenvalues of $B$, $C$ coincide up to sign.