Consider real matrices.
In wikipedia we have a statement about symmetric matrix.
"Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix"
For skew-symmetric matrices is also presented real form, but block diagonal one (denote it as $K_1$),
Now the entries of this form can be presented via appropriate permutations in the form $K_2=PK_1P^T$ where they are grouped on anti-diagonal
for example:
$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & -a & 0 & 0 \\ a & 0 & 0 & 0\\ 0 & 0 & 0 & -b \\ 0 & 0 & b & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \end{bmatrix}= \begin{bmatrix} 0 & 0 & 0 & b\\ 0 & 0 & a & 0 \\ 0 & -a & 0 & 0\\ -b & 0 & 0 & 0 \end{bmatrix} $
hence:
- is it reasonable to add a comment in wikipedia for a sake of analogy with symmetric matrix entry:
"Every skew-symmetric matrix is thus, up to choice of an orthonormal basis, a anti-diagonal (skew-symmetric) matrix" ?
What do you think ? Would be such statement useful to the degree that it is recommended to add it to Wikipedia ? Has it some advantages which are not immediately visible in block diagonal form?