Linear Algebra Done Right · Problem 7D.18

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

A square matrix

A

is called symmetric if it equals its transpose. Prove that if

A

is a symmetric matrix with real entries, then there exists a unitary matrix

Q

with real entries such that

Q^*AQ

is a diagonal matrix.