What is the set $\{A \,\,\mid \text{ any two eigenvectors of }$A$\text{from different eigenspaces are orthogonal.}\}$?

I am reading "Masahiko Saito's Linear Algebra" (in Japanese) by Masahiko Saito.

The following three propositions are in this book.

Let $A$ be a unitary matrix. Then, any two eigenvectors of $A$ from different eigenspaces are orthogonal.

Let $A$ be an Hermitian matrix. Then, any two eigenvectors of $A$ from different eigenspaces are orthogonal.

Let $A$ be a skew-Hermitian matrix. Then, any two eigenvectors of $A$ from different eigenspaces are orthogonal.

What is the set $\{A \,\,\mid \text{ any two eigenvectors of } A \text{ from different eigenspaces are orthogonal }\}$?

Is there a matrix $A\in\{A \,\mid \text{ any two eigenvectors of } A \text{ from different eigenspaces are orthogonal }\}$ which is not a normal matrix?
If so, please give me an example.

Solution 1:

Hyperplane, Thank you very much for your answer.

Let $A=\begin{pmatrix}1&1&0\\0&1&0\\0&0&0\end{pmatrix}$.
Then $AA^{*}=\begin{pmatrix}2&1&0\\1&1&0\\0&0&0\end{pmatrix}$.
Then $A^{*}A=\begin{pmatrix}1&1&0\\1&2&0\\0&0&0\end{pmatrix}$.
So, $A$ is not a normal matrix.
The eigenvalues of $A$ are $1$ and $0$.
The eigenspace of $A$ corresponding to $1$ is $\operatorname{Nul}\begin{pmatrix}0&1&0\\0&0&0\\0&0&-1\end{pmatrix}=\Biggr\{\alpha\begin{pmatrix}1\\0\\0\end{pmatrix} \mid\alpha\in\mathbb{C}\Biggl\}$.
The eigenspace of $A$ corresponding to $0$ is $\operatorname{Nul}\begin{pmatrix}1&1&0\\0&1&0\\0&0&0\end{pmatrix}=\Biggr\{\beta\begin{pmatrix}0\\0\\1\end{pmatrix} \mid\beta\in\mathbb{C}\Biggl\}$.
So, any two eigenvectors of $A$ from different eigenspaces are orthogonal.