$U \in M_n$ is **unitary** (酉矩阵) if $U^∗ U = I$.
If $U$ is real, usually say $U$ **orthogonal**.

A set of vectors $\{x_1, \dots, x_n\}$ is an **orthogonal set** if $x_i^∗ x_j = 0 \forall i \ne j$.
If $\|x_i\| = 1, i = 1, \dots, n$, then we call it **orthonormal** (标准正交).

Property:
An orthogonal set of nonzero vectors is linearly independent.

Theorem: TFAE:

- $U$ is unitary;
- $U$ nonsingular and $inv(U) = U^∗$;
- $U U^∗ = I$;
- $U^∗$ is unitary;
- Columns of $U$ are orthonormal;
- Rows of $U^∗$ orthonormal;
- $U$ is an isometry: $(U x)^∗ (U x) = x^∗ x, \forall x$

**Schur’s Lemma**:
Given $A \in M_n$, with eigenvalues $\lambda_1, \dots, \lambda_n$ in any order,
then exist unitary $U$ s.t. $U^∗ A U = T$,
with $T$ upper-diagonal with diagonal elements the eigenvalues.
If $A$ real and all its eigenvalues real, then $U$ can be real and orthogonal.

Theorem:
Let $F$ be a commuting family, then exist unitary $U$ s.t. $U^∗ A U$ triangular for any $A \in F$.
If $F$ is real with real eigenvalues, we can do it over $\mathbb{R}$,
with diagonal blocks 1-by-1 or 2-by-2.

## Normal Matrix

A matrix is **normal** (正规) if it commutes with its adjoint (共轭转置).

(Note: Similarity does not guarantee same eigenvectors.)

Theorem (**Spectral theorem for normal matrices**):
Let $A in M_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, TFAE:

- A is normal;
- A is unitarily diagonalizable;
- $\Sigma |a_{ij}|^2 = \Sigma |\lambda_i|^2$
- exist orthonormal set of n eigenvectors of A.

Similarity transformations:

- A arbitrary:
$M^{-1} A M = J$, with columns of M eigenvectors & "generalized eigenvectors", and J Jordan form.
- A diagonalizable: $S^{-1} A S = \Lambda$, with columns of S eigenvectors and $\Lambda$ diagonal.

**Schur’s Lemma**:

- A arbitrary: exist unitary U s.t. $U^∗ A U = T$, with T upper triangular.
- A normal: exist unitary U s.t. $U^∗ A U = \Lambda$, with $\Lambda$ diagonal.

Theorem (Proof not shown):
Every matrix is similar to a symmetric matrix.
("Symmetric Jordan form")

Special cases of normal matrices:

- Hermitian: $\Lambda$ is real.
- Real symmetric: $\Lambda$ is real, and U is (real) orthogonal matrix.

- Skew hermitian: $\Lambda$ is pure imagery;
- Unitary: norm of eigenvalue = 1;

🏷 Category=Algebra Category=Matrix Theory