Polar decompositions of quaternion matrices in indefinite inner product spaces

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process, an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.

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Structure-preserving Diagonalization of Matrices in Indefinite Inner Product Spaces

Electronic Journal of Linear Algebra ◽

10.13001/ela.2020.5071 ◽

2020 ◽

Vol 36 (36) ◽

pp. 21-37

Author(s):

Philip Saltenberger

Keyword(s):

Sufficient Conditions ◽

Inner Product ◽

Product Spaces ◽

Hermitian Matrices ◽

Additive Decomposition ◽

Indefinite Inner Product ◽

Structure Preserving ◽

Inner Product Spaces ◽

Structured Matrix ◽

Necessary And Sufficient

In this work some results on the structure-preserving diagonalization of selfadjoint and skewadjoint matrices in indefinite inner product spaces are presented. In particular, necessary and sufficient conditions on the symplectic diagonalizability of (skew)-Hamiltonian matrices and the perplectic diagonalizability of per(skew)-Hermitian matrices are provided. Assuming the structured matrix at hand is additionally normal, it is shown that any symplectic or perplectic diagonalization can always be constructed to be unitary. As a consequence of this fact, the existence of a unitary, structure-preserving diagonalization is equivalent to the existence of a specially structured additive decomposition of such matrices. The implications of this decomposition are illustrated by several examples.

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The Action of the Weyl Group on the $$E_8$$ Root System

Graphs and Combinatorics ◽

10.1007/s00373-021-02315-8 ◽

2021 ◽

Author(s):

Rosa Winter ◽

Ronald van Luijk

Keyword(s):

Automorphism Group ◽

Weyl Group ◽

Root System ◽

Sufficient Conditions ◽

Maximal Clique ◽

Necessary And Sufficient Conditions ◽

Inner Product ◽

Colored Graphs ◽

Necessary And Sufficient ◽

Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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