$U \in M_n$ 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:

1. $U$ is unitary;
2. $U$ nonsingular and $inv(U) = U^∗$;
3. $U U^∗ = I$;
4. $U^∗$ is unitary;
5. Columns of $U$ are orthonormal;
6. Rows of $U^∗$ orthonormal;
7. $U$ 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:

1. A is normal;
2. A is unitarily diagonalizable;
3. $\Sigma |a_{ij}|^2 = \Sigma |\lambda_i|^2$
4. 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;