Lemma. Let be a self-adjoint operator on a finite-dimensional inner product space . Then
- Every eigenvalue of is real.
- Suppose that is a real inner product space. Then the characteristic polynomial of splits.
Theorem (Spectral Theorem for self-adjoint operator). Let be a linear operator on a finite-dimensional inner product space over . Then is self-adjoint if and only if there exists an orthonormal basis for consisting of eigenvectors of .
Definition. We say a symmetric real matrix is said to be positive definite if is positive for every non-zero column vector of real numbers.
It is a quite easy consequence that if is positive definite symmetric matrix with real coefficients, all its eigenvalues are positive.
Exercise. Let be a positive definite symmetric matrix with
real coefficients. Show that
This generalizes the fact that , which corresponds to the case where is the identity.
Proof. Since is positive definite symmetric matrix with real coefficients, by spectral theorem, write where is a rotation
and is diagonal with entries , where are the eigenvalues of . Then we have
Since for , we have