Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(a_{ij}^\sigma)$, where $A = (a_{ij})$, and the $T$-exponent means "transposed of."
Example: $k = \mathbb{C}$ and $\sigma$ is complex conjugation. In that case, $A$ is called a Hermitian matrix.
Question: Is there a spectral theorem known which naturally generalizes the spectral theorem for complex Hermitian matrices?
What about if $k$ is algebraically closed, or real-closed?
