Abstract
This chapter examines the statistical properties of random matrices whose probability density belongs to the class of matrix radial densities. The results presented in this chapter constitute the theoretical foundations of the algorithms for generation of random matrices uniformly distributed in norm bounded sets subsequently presented in Chap. 18.
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Notes
- 1.
The number \(n_{v}=nm-\frac{n}{2}(n+1)\) of free variables needed to represent an m×n matrix V=[v 1 ⋯ v n ], m≥n with orthonormal columns can be constructed as follows: the first column v 1 can be chosen in m−1 different ways (m free variables with one norm constraint), v 2 can be chosen in m−2 different ways (m free variables with one norm constraint and one orthogonality constraint), …,v n can be chosen in m−n different ways.
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Tempo, R., Calafiore, G., Dabbene, F. (2013). Statistical Theory of Random Matrices. In: Randomized Algorithms for Analysis and Control of Uncertain Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-4610-0_17
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