A Survey Of Canonical Forms And Invariants For Unitary Similarity

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Linear Algebra And Its Applications


Matrices A and B are said to be unitarily similar if U*AU = B for some unitary matrix U. This expository paper surveys results on canonical forms and invariants for unitary similarity. The first half gives a detailed description of methods developed by several authors (Brenner, Littlewood, Mitchell, McRae, Radjavi, Sergeĭchuk, and Benedetti and Cragnolini) using inductively defined reduction procedures to transform matrices to canonical form. The matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form. At each stage, one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks. The process ends in a finite number of steps, producing both the canonical form and the subgroup of the unitary group that preserves that form. Depending on the initial step, various canonical forms may be defined. The method can also be used to define canonical forms relative to certain subgroups of the unitary group, and canonical forms for finite sets of matrices under simultaneous unitary similarity. The remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity, such as the Specht-Pearcy trace invariants, the numerical range, and unitary reducibility.