Let $W$ be a square matrix. If $W+W'$ is positive definite, what one can say about $W$?

Question

Let $W$ be a square matrix. If $W+W'$ is positive definite, what one can say about $W$?

Can you say that $W$ is invertible?

Can you say that $W$ is positive or negative definite?

Notation: $'$ means tranpose

Answer 1

Edit: Assuming $W$ is symmetric. Suppose $W+W^T$ is positive definite and let $v$ be an eigenvector of $W$ with corresponding eigenvalue $\lambda$, then we see that \begin{align} 2\lambda || v||^2=v^T(W+W^T)v > 0 \end{align} which means $\lambda > 0$. Thus, if $W+W^T$ is positive definite, then $W$ and, of course, $W^T$, will also be positive definite (Note: $W$ is invertible since $\lambda>0$).

Answer 2

They may even fail to be positive or negative semidefinite: $$ \begin{bmatrix}2&1\\1&1\end{bmatrix}=\begin{bmatrix}2&1\\0&0\end{bmatrix}+\begin{bmatrix}0&0\\1&1\end{bmatrix} $$

Edit: after the question has been edited to mean something else. If $W$ is positive-definite, then $W'$ is also positive-definite (assuming we are talking real matrices), because $$x'W'x=(x'W'x)'=x'Wx>0,$$ and then $W+W'$ is positive definite since the sum of positive-definite is again positive-definite.

Answer 3

Let $W$ be the 3x3 matrix with ones on the diagonal and $0.1$ above the diagonal. Then, $W$ is not positive definite (or negative definite) since it isn't even symmetric. But, $W^T + W$ is positive definite.