Perspectives on the Spectral Theorem

Abstract

Suppose A is a self-adjoint n × n matrix, meaning that $A_{kj} = \overline{A_{jk}}$ for all 1≤ j, k ≤ n. Then a standard result in linear algebra asserts that there exist an orthonormal basis $\{\mathbf{v}_{j}\}_{j=1}^{n}$ for ${\mathbb{C}}^{n}$ and real numbers λ ₁,…,λ _n such that $A\mathbf{v}_{j} =\lambda _{j}\mathbf{v}_{j}$. (See Theorem 18 in Chap. 8 of [24] and Exercise 4 in Chap. 7)