Similarity transformations are viewed as alternative interpretations of a matrix operator; the associated theorems address its basis-free descriptors. Diagonalization is heralded as a holy grail, facilitating scads of algebraic manipulations such as inversion, root extraction, and power series evaluation. Schur decomposition, though ponderous, provides a valuable instrument for understanding the orthogonal diagonalizability of normal matrices, as well as the Cayley–Hamilton theorem. A physical experiment illustrating the instability of principal axis rotations is employed to stimulate insight into quadratic forms. The chapter also provides a transparent exposition of the properties and applications of the singular value decomposition, including rank reduction and the pseudoinverse. The practical futility of eigenvector calculation through the characteristic polynomial is outlined in a section devoted to a bird’s-eye perspective of the QR algorithm. The role of randomness in its numerical implementation, as well as in the occurrence of defective matrices, is addressed. Projects address positive definite matrices, Hessenberg forms, the discrete Fourier transform, and advanced aspects of the singular value decomposition.