Vector space/Inner product/Endomorphism/Self-adjoint/Eigentheory/Fact

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let

$\varphi \colon V\longrightarrow V$

be a self-adjoint endomorphism. Then the following statements hold.

1. For a ${}\varphi$-invariant linear subspace ${}U\subseteq V$, also the orthogonal complement ${}U^{\perp }$ is ${}\varphi$-invariant.
2. All eigenvalues are real.
3. The eigenspaces for different eigenvalues are orthogonal to each other.
4. Let ${}V$ be finite-dimensional. Then the characteristic polynomial of ${}\varphi$ splits into linear factors.

Proof, Write another proof