Canonical form for orthogonal similarity classes

Question

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar?

That is, canonical representatives for the equivalence class defined by $A\sim B$ iff $\exists\, U$ unitary such that $A = UBU^*$.

Answer 1

Here's a reference:

Shapiro, H., A survey of canonical forms and invariants for unitary similarity, Linear Algebra Appl. 147:101–168 (1991).

I quote from the introduction:

Many authors have studied the problem of finding a canonical form for unitary similarity and proposed methods for reducing a matrix to a canonical form under unitary similarity. As we shall see, these reduction methods are based on inductive procedures that involve partitioning the matrix into blocks and successively applying unitary transformations to reduce the blocks to some nice form. The final "canonical form" is usually not easily visualized.

For $2\times 2$ matrices, the canonical form is given by Theorem 2.4 in the paper: every matrix is unitarily similar to an upper triangular matrix with nonnegative entry in the upper right block, and to only one such (modulo permutation of the diagonal entries).

One can get rid of "modulo permutations" by introducing some order on $\mathbb C^2$, such as lexicographic order. This isn't particularly elegant, but this sort of thing is inherent in the problem. "What is a canonical form of vectors in $\mathbb C^n$ modulo permutation of coordinates?" is a special case of this question (diagonal matrices).

The canonical forms for larger matrices are much messier.

Answer 2

This problem contains the problem of classifying an arbitrary system of unitary spaces (i.e., complex inner product spaces) and linear mappings between these spaces; that is, the problem of classifying unitary representations of any quiver. The problem that contains the problem of classifying matrices under unitary similarity is called unitarily wild. Nevertheless, Littlewood's algorithm reduces each square complex matrix A to some matrix A' that is unitarily similar to A such that A and B are unitarily similar iff A'=B'. The matrix A' can be considered as canonical for A. See details and some description of canonical matrices in [V.V. Sergeichuk, Unitary and Euclidean representations of a quiver, Linear Algebra Appl. 278 (1998) 37-62].