Self-adjoint: the real-symmetric heroes
An operator is self-adjoint when T* = T (Hermitian over C, symmetric over R). The defining symmetry <Tv, w> = <v, Tw> forces two beautiful consequences: every eigenvalue is real, and eigenvectors for distinct eigenvalues are orthogonal. These operators are the abstract observables of quantum mechanics and the curvature/inertia matrices of physics.
A self-adjoint T is positive (semi)definite when <Tv, v> >= 0 for all v — equivalently, all its eigenvalues are nonnegative. These are the operators with square roots, the ones that act like "squared lengths", and exactly the generalization of Vol I's positive definite matrices.
Unitary: the rigid motions
An isometry preserves length: ||Tv|| = ||v|| for all v, and hence preserves every inner product and angle. When such a map is also onto (automatic in finite dimensions) it is a unitary operator, characterized by T* T = T T* = I — equivalently T* = T^-1. Over R the same condition names the orthogonal matrices: rotations and reflections.
Unitary operators are the symmetries of an inner product space — they shuffle vectors without distorting geometry. Their eigenvalues all sit on the unit circle (|lambda| = 1), and their columns form an orthonormal basis. They are the change-of-basis maps that keep everything orthonormal.
Normal: the unifying class
Self-adjoint and unitary operators look different, yet both satisfy one shared equation: T commutes with its own adjoint, T* T = T T*. An operator obeying this is normal. Normality is the exact umbrella that contains self-adjoint (T* = T), unitary (T* = T^-1), and positive operators all at once.
The classification at a glance (all assume an inner product space): self-adjoint T* = T real eigenvalues positive <Tv,v> >= 0 eigenvalues >= 0 (subset of self-adjoint) unitary T* = T^-1 |eigenvalue| = 1 isometry ||Tv|| = ||v|| (= unitary in finite dim) normal T*T = T T* the umbrella over ALL of the above Test of normality for T with matrix [0, -1; 1, 0] (a 90-deg rotation): T* = T^T = [0, 1; -1, 0] T*T = [1, 0; 0, 1] = I T T*= [1, 0; 0, 1] = I equal -> T is normal (in fact unitary).
Why does normality earn a name of its own? Because it is the precise dividing line for diagonalizability by an orthonormal basis. The complex spectral theorem states: T is unitarily diagonalizable if and only if T is normal. Every operator in this guide can therefore be rotated into a diagonal of pure eigenvalues — geometry handed back in its cleanest form. That theorem, and its real symmetric counterpart, is the payoff the whole volume has been building toward.