# Generalization of integration of e^{-x^2}

By | August 14, 2015

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 