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Matrices and Their (Un)Faithful Fermi-quadratic Forms

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Abstract

We consider the algebra of n × n matrices over ℂ and their corresponding Fermi-quadratic forms. The properties of these operators are studied with respect to the properties of the underlying matrices. It is well known that these Fermi-quadratic forms have a faithful matrix representation. The purpose of this article is to investigate the (un)faithful representation of the matrix algebra by its Fermi-quadratic forms. The preservation of the matrix commutators, anticommutators, and eigenvalues in the Fermi-quadratic forms are discussed. Other matrix functions such as the exponential function are studied, as well as an application to quantum channels where we consider density matrices and operators and the Kraus representation. Lastly, we consider extensions of these quadratic forms and entangled states that arise from these forms.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

    Yorick Hardy

  2. International School for Scientific Computing, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa

    Willi-Hans Steeb

  3. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa

    Garreth Kemp

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  1. Yorick Hardy
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  2. Willi-Hans Steeb
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  3. Garreth Kemp
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Correspondence to Yorick Hardy.

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Hardy, Y., Steeb, WH. & Kemp, G. Matrices and Their (Un)Faithful Fermi-quadratic Forms. Int J Theor Phys 58, 1470–1481 (2019). https://doi.org/10.1007/s10773-019-04034-6

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  • DOI: https://doi.org/10.1007/s10773-019-04034-6

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