bibkey: garcia2017partial
authors: Stephan Ramon Garcia, David Sherman
year: 2017
title: Matrices Similar to Partial Isometries
doi: null
url: https://arxiv.org/abs/1703.02960v2
claim: Lemma 2 constructs finite partial isometries with zero and prescribed open-unit-disk eigenvalues; Theorem 1 characterizes similarity to partial isometries using spectrum, unit-circle semisimplicity and kernel dimensions.
strata_touched: []
license: citation-only
triage: anchor

Prescribed spectra of partial isometries

The cited version is arXiv:1703.02960v2, posted 15 March 2017; its manuscript prints 16 March 2017. Lemma 2 constructs an upper triangular partial isometry with diagonal zero followed by the prescribed open-unit-disk eigenvalues, orthonormal nonzero columns and nonzero first-superdiagonal entries. Theorem 1 characterizes similarity, not unitary equivalence: the spectrum lies in the closed disk, unit-circle eigenvalues are semisimple, and the nullity at zero bounds the geometric multiplicity of every open-disk eigenvalue.

These are classical matrix tools used within the shared-processor construction in quantum-volume §213.4. They do not establish that the channel-specific sharp three-dimensional models embed into one processor, nor that one preparation curve attains sharp calibration at several prescribed points. The former is proved there by the normalized eigenvector Gram identity; the latter retains the distinct lower and upper bounds of §213.5. No general similarity theorem is being promoted to unitary equivalence without that additional argument.

Section 217 uses these same stable Gram models in orthogonal layers to realize prescribed eigenvalue multiplicities at several sharp locations. The resulting mixed preparation occupies that common space; geometric multiplicity lower bounds come from the channel-specific cross-Gram argument, not from a similarity-to-partial-isometry statement. Sections 215 and 217 settle the corresponding mixed single-curve minima; the general pure single-curve problem retains its separate scope.