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Non-asymptotic, Local Theory of Random Matrices

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Abstract

Chapters 4 and 5 are the core of this book. Chapter 6 is included to form the comparison with this chapter. The development of the theory in this chapter will culminate in the sense of random matrices. The point of viewing this chapter as a novel statistical tool will have far-reaching impact on applications such as covariance matrix estimation, detection, compressed sensing, low-rank matrix recovery, etc. Two primary examples are: (1) approximation of covariance matrix; (2) restricted isometry property (see Chap. 7).

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Notes

  1. 1.

    Let \(\mathcal{A}\) is a linear transformation from vector V to vector W. The subset in V

    $$\displaystyle{\ker (\mathcal{A}) = \left \{v \in V: \mathcal{A}(v) = 0 \in W\right \}}$$

    is a subspace of V, called the kernel or null space of A. 

  2. 2.

    More generally, in this section we estimate the second moment matrix \(\mathbb{E}\left (\mathbf{x} \otimes \mathbf{x}\right )\) of an arbitrary random vector x (not necessarily centered).

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  1. Tennessee Technological University, Cookeville, Tennessee, USA

    Robert Qiu

  2. Utica, NY, USA

    Michael Wicks

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Qiu, R., Wicks, M. (2014). Non-asymptotic, Local Theory of Random Matrices. In: Cognitive Networked Sensing and Big Data. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4544-9_5

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