A Hermitian matrix \(A\in M_n(\CC)\) has real eigenvalues and an orthonormal basis of eigenvectors. By the Courant–Fischer theorem, its eigenvalues \(\lambda_1\ge\cdots\ge\lambda_n\) are min-max values of the Rayleigh quotient \(x^*Ax/x^*x\) over subspaces of \(\CC^n\), and the interlacing theorems place the eigenvalues of a principal submatrix of \(A\), and of a rank-one Hermitian perturbation of \(A\), between consecutive eigenvalues of \(A\). A matrix is unitarily diagonalizable if and only if it is normal. Every \(A\in M_{m\times n}(\CC)\) has a singular value decomposition \(A=U\Sigma V^*\) with \(U\) and \(V\) unitary and \(\Sigma\) diagonal with nonnegative entries. A positive semidefinite matrix has a unique positive semidefinite square root, and every square matrix has a polar decomposition \(A=UH\) with \(U\) unitary and \(H=(A^*A)^{1/2}\). The Jordan form of a nilpotent matrix \(N\) is determined by the ranks of the powers \(N^k\), and the Jordan form does not depend continuously on the matrix: \(\begin{pmatrix}0&\varepsilon\\0&0\end{pmatrix}\) has one Jordan block for \(\varepsilon\ne0\) and two for \(\varepsilon=0\).