Unitary matrix (酉矩阵) is a square matrix whose inverse is its Hermitian adjoint (共轭转置): $U \in M_n$, $U^H U = I_n$. Real unitary matrices are usually called orthogonal matrices. A set of vectors $\{x_1, \dots, x_n\}$ is an orthogonal set if $\forall i \ne j, x_i^H x_j = 0$. It is orthonormal (标准正交) if, in addition, $\forall i, \|x_i\| = 1$. An orthogonal set of nonzero vectors is linearly independent.
Theorem: TFAE:
Schur's Lemma (unitary triangularization): Any square matrix can be unitarily triangularized with its eigenvalues on the diagonal: $\forall A \in M_n$, $\exists U \in U(n)$, $T \in T(n)$ (upper triangular), $\text{diag}(A) = \lambda(A)$: $U^H A U = T$. Any square real matrix with real eigenvalues can be orthogonally triangularized with its eigenvalues on the diagonal.
Theorem: Any communting family of square matrices can be simultaenously unitarily triangularized: $\mathcal{F} \subset M_n$, $\forall A, B \in \mathcal{F}$, $A B = B A$, $\exists U \in U(n)$: $\forall A \in \mathcal{F}$, $U^H A U \in T(n)$ (upper triangular). Any communting family of real square matrices with real eigenvalues, can be simultaenously orthogonally block triangularized, with diagonal blocks 1-by-1 or 2-by-2.
For an arbitrary square matrix $A$, there exist $M$ whose columns are eigenvectors and "generalized eigenvectors" of $A$, and $J$ a Jordan form, such that $M^{-1} A M = J$. If $A$ is diagonable, then for any basis $S$ whose columns are eigenvectors of $A$, we have $S^{-1} A S = \Lambda$ where $\Lambda$ is a diagonal matrix with the eigenvalues of $A$ on the diagonal.
(Note: Similarity transforms preserve eigenvalues but not eigenvectors.)
Theorem ("Symmetric Jordan form"; proof not shown): Every matrix is similar to a symmetric matrix.
Normal matrix (正规矩阵) is a square matrix that commutes with its Hermitian adjoint: $A \in M_n$, $A A^H = A^H A$; or equivalently (see the spetral theorem below), a square matrix that is unitarily diagonalizable: $\exists U \in U(n)$, $\lambda \in \mathbb{C}^n$: $A = U \Lambda U^H$. Comparing with Schur's Lemma, normal matrices are exactly those square matrices that can be unitarily diagonalized, instead of just being triangularized.
Spectral theorem for normal matrices: Let $A \in M_n$, TFAE: (1) $A$ is normal; (2) $A$ is unitarily diagonalizable; (3) There exists an orthonormal set of $n$ eigenvectors of $A$. (4) $\|A\|_F = \|\lambda(A)\|$, that is, $\sum_{i,j=1}^n |a_{ij}|^2 = \sum_{i=1}^n |\lambda_i|^2$.
Special cases of normal matrices as identified by their eigenvalues: (1) Hermitian, $\lambda \subset \mathbb{R}$; (2) skew-Hermitian, $\lambda \subset i \mathbb{R}$; (3) unitary, $\lambda \subset e^{i \mathbb{R}}$. All normal matrices have unitary bases of eigenvectors, but the real versions of these special cases (symmetric, skew-symmetric, and orthogonal) can have orthogonal bases of eigenvectors.