Norms on complex matrices induced by random vectors

Abstract

We introduce a family of norms on the 𝑛 × 𝑛 complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.

Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms

“…This provides one way to visualize the properties of random vector norms. We consider a few examples hereand refer the reader to [7, Section 2] for further examples and details.…”

“…Suppose is an even integer and is a random vector whose entries are independent normal random variables with mean and variance . The example in [7, equation (2.12)] illustrates in which is the Frobenius norm. For , the extension to guaranteed by Theorem 1.3 is [7, p. 816].…”

“…Lewis proved that any weakly unitarily invariant norm on H n is a symmetric vector norm applied to the eigenvalues [11,Section 8]. 448 Á. Chávez, S. R. Garcia, and J. Hurley Our first result is a short proof of Lewis' theorem that avoids his theory of group invariance in convex matrix analysis [11], the wonderful but complicated framework that underpins [1,7]. Our new approach uses more standard techniques, such as Birkhoff 's theorem on doubly stochastic matrices [6].…”

“…The random-vector norms of the next theorem are weakly unitarily invariant norms on H n that extend to weakly unitarily invariant norms on M n (see Theorem 1.3). They appeared in [7], and they generalize the complete homogeneous symmetric polynomial norms of [1, Theorem 1]. The original proof of [7, Theorem 1.1(a)] requires d ≥ 2 and relies heavily on Lewis' framework for group invariance in convex matrix analysis [11].…”

“…The simplified proof of Theorem 1.1 and the extension of Theorem 1.2 from d ≥ 2 to d ≥ 1 permit the main results of [7], restated below as Theorem 1.3, to rest on simpler foundations while enjoying a wider range of applicability. The many perspectives offered in Theorem 1.3 explain the normalization in (1.1).…”