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 bepositive definiteif 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