
Contents
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3.1 finite dimensional quaternion vector spaces 3.1 finite dimensional quaternion vector spaces
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3.2 Matrix algebra 3.2 Matrix algebra
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3.3 Real matrix representation of quaternions 3.3 Real matrix representation of quaternions
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3.4 Complex matrix representation of quaternions 3.4 Complex matrix representation of quaternions
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3.5 Numerical ranges with respect to conjugation 3.5 Numerical ranges with respect to conjugation
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3.6 Matrix decompositions: nonstandard involutions 3.6 Matrix decompositions: nonstandard involutions
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3.7 Numerical ranges with respect to nonstandard involutions 3.7 Numerical ranges with respect to nonstandard involutions
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3.8 Proof of Theorem 3.7.5 3.8 Proof of Theorem 3.7.5
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3.9 The metric space of subspaces 3.9 The metric space of subspaces
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3.10 Appendix: Multivariable real analysis 3.10 Appendix: Multivariable real analysis
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3.11 Exercises 3.11 Exercises
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3.12 Notes 3.12 Notes
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Three Vector spaces and matrices: Basic theory
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Published:August 2014
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Abstract
This chapter covers the basics on the vector space of columns with quaternion components, matrix algebra, and various matrix decompositions. The real and complex representations of quaternions are extended to vectors and matrices. Various matrix decompositions are studied; in particular, Cholesky factorization is proved for matrices that are hermitian with respect to involutions other than the conjugation. A large part of this chapter is devoted to numerical ranges of quaternion matrices with respect to conjugation as well as with respect to other involutions. Finally, a brief exposition is given for the set of quaternion subspaces, understood as a metric space with respect to the metric induced by the gap function.
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