Handbook of Linear Algebra, 2nd Edition

Chapter 8

Unitary Similarity, Normal Matrices, and Spectral Theory

Helene Shapiro

Swarthmore College

Unitary transformations preserve the inner product. Hence, they preserve metric quantities that stem from the inner product, such as length, distance, and angle. While a general similarity preserves algebraic features of a linear transformation, such as the characteristic and minimal polynomials, the rank, and the Jordan canonical form, unitary similarities also preserve metric features such as the norm, singular values, and the numerical range. Unitary similarities are desirable in computational linear algebra for stability reasons.

Normal transformations are those that have an orthogonal basis of eigenvectors and, thus, can be represented by ...