
Published online:
19 October 2017
Published in print:
24 August 2014
Online ISBN:
9781400852741
Print ISBN:
9780691161853
Contents
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12.1 Canonical forms for ϕ-mixed pencils: Strict equivalence 12.1 Canonical forms for ϕ-mixed pencils: Strict equivalence
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12.2 Proof of Theorem 12.1.2 12.2 Proof of Theorem 12.1.2
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12.3 Canonical forms of ϕ-mixed pencils: Congruence 12.3 Canonical forms of ϕ-mixed pencils: Congruence
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12.4 Proof of Theorem 12.3.1 12.4 Proof of Theorem 12.3.1
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12.5 Strict equivalence versus ϕ-Congruence 12.5 Strict equivalence versus ϕ-Congruence
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12.6 Canonical forms of matrices under ϕ-congruence 12.6 Canonical forms of matrices under ϕ-congruence
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12.7 Comparison with real and complex matrices 12.7 Comparison with real and complex matrices
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12.8 Proof of Theorem 12.7.4 12.8 Proof of Theorem 12.7.4
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12.9 Exercises 12.9 Exercises
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12.10 Notes 12.10 Notes
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Chapter
Twelve Mixed matrix pencils: Nonstandard involutions
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Published:August 2014
Cite
Rodman, Leiba, 'Mixed matrix pencils: Nonstandard involutions', Topics in Quaternion Linear Algebra (Princeton, NJ , 2014; online edn, Princeton Scholarship Online, 19 Oct. 2017), https://doi.org/10.23943/princeton/9780691161853.003.0012, accessed 23 Apr. 2025.
Abstract
This chapter also studies the canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is φ-hermitian and the other is φ-skewhermitian, with respect to simultaneous φ-congruence. It starts by formulating the canonical form for φ-hsk matrix pencils under strict equivalence. Other canonical forms of mixed matrix pencils are developed with respect to strict equivalence. As an application, this chapter provides canonical forms of quaternion matrices under φ-congruence. As in the preceding chapter, this chapter also fixes a nonstandard involution φ throughout and a quaternion β(φ) such that φ=(β(φ)) = −β(φ) and ∣β(φ)∣ = 1.
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