= Solution
{c}
For a <self-adjoint operator> on a finite-dimensional complex <inner product space>, every eigenvalue is real and eigenspaces for distinct eigenvalues are orthogonal. The <spectral theorem> gives an orthonormal eigenbasis. Accordingly
$$
\boxed{V=V_1\mathbin{\perp}\cdots\mathbin{\perp}V_r,}
$$
and the $E_i$ are orthogonal projections satisfying $E_i^*=E_i$. Since $T$ is nonsingular, none of its eigenvalues is zero. Its representing matrix in the orthonormal eigenbasis is real diagonal.
Back to article page
