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A sign pattern is an array with entries in $$\{+,-,0\}$$. A real matrix $$Q$$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A. Curtis and B.L. Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228-259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $$5\times n$$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP.
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This content will become publicly available on January 1, 2026
The non-symmetric strong multiplicity property for sign patterns
We develop a non-symmetric strong multiplicity property for matrices that may or may not be symmetric. We say a sign pattern allows the non-symmetric strong multiplicity property if there is a matrix with the non-symmetric strong multiplicity property that has the given sign pattern. We show that this property of a matrix pattern preserves multiplicities of eigenvalues for superpatterns of the pattern. We also provide a bifurcation lemma, showing that a matrix pattern with the property also allows refinements of the multiplicity list of eigenvalues. We conclude by demonstrating how this property can help with the inverse eigenvalue problem of determining the number of distinct eigenvalues allowed by a sign pattern.
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- Award ID(s):
- 1839918
- PAR ID:
- 10582824
- Publisher / Repository:
- International Linear Algebra Society
- Date Published:
- Journal Name:
- The Electronic Journal of Linear Algebra
- Volume:
- 41
- ISSN:
- 1081-3810
- Page Range / eLocation ID:
- 153 to 165
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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