27 - Orthonormal Basis for Self-Adjoint Maps

Orthonormal Basis for Self-Adjoint Maps

Let $T:V\to V$ be a linear self-adjoint map and $V$ be a finite dimensional vector space then

There exists orthonormal basis of eigenvectors for $T$

Proof

By Eigenvalues of Adjoint Maps being real then there exists $\lambda \in \mathbb{R}$ and $v\ne 0$ such that

$$T(v)+=\lambda v$$

Consider $U=⟨v⟩$ then $U$ is $T$-invariant and by Orthogonal Complements of Invariant Subspaces then

$$T{\mid }_{{⟨V⟩}^{\perp }}:⟨v{⟩}^{\perp }\to ⟨v{⟩}^{\perp }$$

is well-defined and self-adjoint

So by induction on $n:=\dim V$ then there exists orthonormal basis set

$$\{e_2,\cdots ,e_n\}\text{ of eigenvectors for }T{\mid }_{{⟨v⟩}^{\perp }}$$

Define $e_1=\frac{v}{||v||}$ then

$$\{e_1,\cdot ,e_n\}\text{ is a orthonormal basis of eigenvectors for }T$$