The sums of symplectic, Hamiltonian, and skew-Hamiltonian matrices

Cruz, Ralph John de la; Paras, Agnes T.

doi:10.1016/j.laa.2020.05.036

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Proof Of Theorem 12 For1

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2022

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“…which is partitioned conformal to XAX −1 . It follows that B 2 1 = A 2 1 + 4I, that is, the nilpotent matrix A 2 1 + 4I has a square root, and this gives the Jordan block restrictions stated in Theorem 1.2(a). This proves necessity.…”

Section: Proof Of Theorem 12 Formentioning

confidence: 94%

Sums of orthogonal, symmetric, and skew-symmetric matrices

Cruz

Paras²

2022

ELA

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An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, and $A^T=A^{-1}$, respectively. We give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type ``"orthogonal $+$ symmetric" in terms of the Jordan form of $A-A^T$. We also give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type "orthogonal $+$ skew-symmetric" in terms of the Jordan form of $A+A^T$.

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Section: Proof Of Theorem 12 Formentioning

confidence: 94%