Question

Let \(A\) be a square real matrix all of whose eigenvalues are zero. Show that \(A\) is diagonalizable (that is, similar to a possibly comples diagonal matrix) if and only if \(A=0\).

Answer 1

If \(A\) is diagonalizable, it can be written as \(P^{-1} D P\) where \(D\) is a diagonal matrix and \(P\) is a non-singular matrix. Since all eigenvalues of \(A\) are zero, the diagonal matrix \(D\) must have only zero entries. Thus, \(A=P^{-1} D P=0\).

Conversely, if \(A=0\), then \(A\) is diagonalizable by any invertible matrix \(P\) (since \(P^{-1} A P=\) \(P^{-1} 0 P=D P=0\) ), so \(A\) is diagonalizable and all its eigenvalues are zero.