Constructing matrix with odd constraints

Question

I have been fishing for square matrices that follow $AA^T+A^TA=2nI$ and $A^2+(A^{T})^2=0$ where $n$ is the order of the matrix using a random matrix generation technique,

amat = RandomChoice[{-1,1},{4,4}]; 
While[amat.Transpose[amat] + Transpose[amat].amat != 
8*IdentityMatrix[4], 
While[MatrixPower[amat, 2] != -MatrixPower[Transpose[amat], 2], 
amat = RandomChoice[{-1,1},{4,4}]]]

which does fine at $n=4$, but tends to become computationally inefficient any larger. What would an actual construction method of these matrices look like?

Answer 1

Presumably $A$ is real. The second equation also shows that $A^2$ is skew-symmetric. So, by a change of orthonormal basis, we may assume that $A^2$ is in its real Jordan form. Now, by multiplying both sides of the first equation by $A$ on the left and also on the right, we see that $A^2$ commutes with $A^T$. So, if $A^2$ is singular, $A^T$ and in turn $A$ must be block-diagonal matrices that have the same zero blocks at the position of the zero block of $A^2$. But then $AA^T+A^TA$ is singular, which violates the first equation.

Thus $A^2$ is non-singular. Since eigenvalues of a non-singular skew-symmetric matrix occur in conjugate pairs, $n$ must be even. Now you may take $$ A=Q\left(\underbrace{(R\oplus\cdots\oplus R)}_{k \text{ copies}}\oplus\underbrace{(R^T\oplus\cdots\oplus R^T)}_{\left(\frac n2-k\right) \text{ copies}}\right)Q^T. $$ where $Q$ is any real orthogonal matrix, $R=\sqrt{\frac n2}\pmatrix{1&-1\\ 1&1}$ and $0\le k\le\frac n2$.