Question

Diagonal elements of a skew hermitian matrix

Answer 1

Properties. The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. ... All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e., on the imaginary axis (the number zero is also considered purely imaginary).

Answer 2

imaginary or 0

A skew Hermitian matrix is one in which the conjugated transposition of a complex matrix form is equal to the negative of what it is.

The diagonal elements' positions in the transpose remain unchanged, hence if the matrix has to be Hermitian, each diagonally item should have been the conjugate of itself. As a result, all of the diagonal cells must be actual.

A skew-Hermitian matrix's eigenvalues are all totally imaginary (or even  zero).