Edited: Relationship between the eigenvalues of $\frac{1}{2}(A + A^t)$ and $(A^tA)^{1/2}$

Question

Let $A$ be a square matrix and let $A^t$ be its transpose. I would appreciate references to results regarding the relationship between the eigenvalues of $\frac{1}{2}(A+A^t)$ and those of $(A^tA)^{1/2}$.

Edit: I edited my question and added an answer to my question.

Answer 1

References:

1. Remark 2.10 in Eigenvalue Bounds on Convergence to Stationarity for Nonreversible Markov Chains, with an Application to the Exclusion Process by J. A. Fill. See also the references there.

2. Lemma on p. 288 of CONVEX AND CONCAVE FUNCTIONS OF SINGULAR VALUES OF MATRIX SUMS by R. C. Thompson.

3. Proposition III.5.1 in Matrix Analysis by Bhatia (This one is from https://en.wikipedia.org/wiki/Singular_value)

Answer 2

Without any condition on $A$, there is no relation. For instance if $$ A=\begin{bmatrix} 0&n\\-n&0\end{bmatrix} $$ with $n\in\mathbb N$, then $A+A^t=0$, so its only eigenvalue is zero. Meanwhile, $$ (A^tA)^{1/2}=\begin{bmatrix} n&0\\0&n\end{bmatrix}, $$ with the single eigenvalue $n$.