More on matrices

Abstract

In this chapter the following topics are considered: (1) the eigenvalue problem; (2) a functional equation for square matrices, and (3) the Berezinian for matrices with linearly dependent matrix elements. We find (1) If the ordinary part of a supermatrix is not degenerate, then it can be diagonalized by a similarity transformation. If it is degenerate, then even if it is hermitian, it can normally not be diagonalized. However, superreal hermitian matrices have two-fold degenerate eigenvalues in the fermionic sector. If this is the only degeneracy, then diagonalization is possible. (2) A differentiable function F of a square matrix obeying the functional equation F.A/F.B/ D F.AB/ vanishes identically, or is a power of the superdeterminant. (3) The Berezinian of matrices with linearly dependent matrix elements is determined.