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Cognitive Networked Sensing and Big Data pp 85–144Cite as

Sums of Matrix-Valued Random Variables

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Abstract

This chapter gives an exhaustive treatment of the line of research for sums of matrix-valued random matrices. We will present eight different derivation methods in this context of matrix Laplace transform method. The emphasis is placed on the methods that will be hopefully useful to some engineering applications. Although powerful, the methods are elementary in nature. It is remarkable that some modern results on matrix completion can be simply derived, by using the framework of sums of matrix-valued random matrices.

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Notes

  1. 1.

    The assumption symmetric matrix is too strong for many applications. Since we often deal with complex entries, the assumption of Hermitian matrix is reasonable. This is the fatal flaw of this version. Otherwise, it is very useful.

  2. 2.

    The finite-dimensional operators and matrices are used interchangeably.

  3. 3.

    The idea of free probability is to make algebra (such as operator algebras C  ∗ -algebra, von Neumann algebras) the foundation of the theory, as opposed to other possible choices of foundations such as sets, measures, categories, etc.

  4. 4.

    In analysis the infimum or greatest lower bound of a subset S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers). For example, \(\inf \left \{1, 2, 3\right \} = 1,\inf \left \{x \in \mathbb{R}, 0 < x < 1\right \} = 0\).

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Authors and Affiliations

  1. Tennessee Technological University, Cookeville, Tennessee, USA

    Robert Qiu

  2. Utica, NY, USA

    Michael Wicks

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Qiu, R., Wicks, M. (2014). Sums of Matrix-Valued Random Variables. In: Cognitive Networked Sensing and Big Data. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4544-9_2

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  • DOI: https://doi.org/10.1007/978-1-4614-4544-9_2

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