Matrices and Their (Un)Faithful Fermi-quadratic Forms

  • Yorick HardyEmail author
  • Willi-Hans Steeb
  • Garreth Kemp


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.


Fermi operators Matrix embedding 



  1. 1.
    Bogolubov, N.N., Bogolubov, N.N.: Introduction to Quantum Statistical Mechanics, 2nd edn. World Scientific, Singapore (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bose, A.: Dynkin’s method of computing the terms of the Baker-Campbell-Hausdorff series. J. Math. Phys. 30(9), 2035–2037 (1989). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Colpa, J.H.P.: Diagonalisation of the quadratic fermion Hamiltonian with a linear part. J. Phys. A Math. Gen. 12(4), 469 (1979). ADSCrossRefGoogle Scholar
  4. 4.
    Cunden, F.D., Maltsev, A., Mezzadri, F.: Density and spacings for the energy levels of quadratic Fermi operators. J. Math. Phys. 58(6), 061902 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kraus, K.: States, Effects, and Operations. Lecture Notes in Physics. Springer, Berlin (1983)Google Scholar
  6. 6.
    Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16(3), 407–466 (1961). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Miszczak, J.A.: Singular value decomposition and matrix reorderings in quantum information theory. Int. J. Modern Phys. C 22(09), 897–918 (2011). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Steeb, W.H.: A comment on trace calculations for fermi systems. Acta Phys. Acad. Sci. Hung. 42(3), 171–177 (1977). MathSciNetCrossRefGoogle Scholar
  9. 9.
    Steeb, W.H., Hardy, Y.: Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra, 2nd edn. World Scientific, Singapore (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Steeb, W.H., Hardy, Y.: Bose, Spin and Fermi Systems: Problems and Solutions. World Scientific, Singapore (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.International School for Scientific ComputingUniversity of JohannesburgJohannesburgSouth Africa
  3. 3.Department of Pure and Applied MathematicsUniversity of JohannesburgJohannesburgSouth Africa

Personalised recommendations