Eigenvalues and eigenvectors of supermatrices (Q1117288)

Let \(\Lambda\) be a Grassmann algebra over the complex numbers, generated by a finite or infinite number of odd elements. Let \(\Lambda_ 0\) be the even part and \(\Lambda_ 1\) the odd part of \(\Lambda\). Let \(M=\left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\) be an \(n\times n\) supermatrix where A and D are square with elements in \(\Lambda_ 0\) and B and C have elements in \(\Lambda_ 1\). The authors define the concepts of (super-)eigenvalue and eigenvector for M and prove that if the eigenvalues of \(\tilde A,\) the body of A, are distinct and if the eigenvalues of \(\tilde B\) are also distinct, there exists an invertible supermatrix U such that \(U^{-1}MU\) is diagonal.