A complete set of unitary invariants for 3×3 complex matrices

Abstract

is a complete set of unitary invariants for any nXn complex matrix A. The author was able to improve this result by demonstrating in [3] that for n fixed but arbitrary, there is always a subset of the above collection containing fewer than 24n2 traces which serves as a complete set of unitary invariants for nXn matrices. It is of interest to discover the sharpness of the above upper bound on the number of traces needed, and thus it was desired to compare the estimate 24n2 with any known results. Unfortunately, even though there is a vast literature on the subject, to the author's knowledge no solution to the problem even for small n has been given, except in the case n =2. Murnaghan [1] showed that for n=2, the traces u(A), o(A2), and o-(A*A), form a complete set of invariants, and he began a study of the case n =3. He did not carry his analysis far enough to solve the problem however and, in fact, his result that six traces suffice in the case that the eigenvalues are distinct is incorrect, as will be shown by an example in ?3. It is the purpose of this paper to complete the analysis of the case n = 3 by demonstrating that there is a collection of nine traces which suffice in this situation. 2. The idea is to treat the different eigenvalue possibilities as separate cases, obtain what turns out to be a canonical form under unitary transformations, and then show that two matrices in this form which possess the same nine appropriate traces are equal. We begin with four lemmas of an elementary nature. Details of the proofs of these lemmas can be found in [1]. For the triangular matrix