Elementary Properties of a Self-Adjoint Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and let $T$ be a linear operator on $V$ that is self-adjoint. Then the following hold.

1. (Real values on the diagonal) For every $x\in V$ the complex number $\langle x,T(x)\rangle$ is a real number.

2. (Real eigenvalues) If $\lambda$ is a complex number and $x\in V$ is an eigenvector of $T$ with eigenvalue $\lambda$, then $\lambda$ is a real number.

3. (Invariance of an orthogonal complement) Let $u\in V$ be a unit vector that is an eigenvector of $T$ with eigenvalue $\lambda$ for some complex number $\lambda$, let $\tilde{u}\in V^{1}$ be the $1$-tuple with component $u$, and let $W$ be the orthogonal complement of the span of $\tilde{u}$, which is a linear subspace of $V$ by claim 1 of The Span of a Finite Family is the Smallest Subspace Containing It and The Orthogonal Complement of a Linear Subspace is a Linear Subspace. Then $T(x)\in W$ for every $x\in W$.

4. (Restriction to an invariant subspace) Let $W$ be a linear subspace of $V$ with $T(x)\in W$ for every $x\in W$. Then the map sending each $x\in W$ to $T(x)$ is a self-adjoint linear operator on $W$, the latter being a complex inner product space by claims 1 and 3 of A Linear Subspace is a Vector Space and Inherits an Inner Product.