POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

Moghadasi, Seyed Reza

doi:10.1017/s0004972711003273

Cited by 5 publications

(10 citation statements)

References 4 publications

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“…Remark 9. In the real case (35) corresponds to a result of Wilks [25]; the method of derivation presented here is that of [17]. For each β = 1, 2 and 4 both (38) and (39) can be found in [21], where the derivation made use the Jacobian for a QR type decomposition; see also [14,15].…”

Section: Moments Of Determinantsmentioning

confidence: 99%

“…It is an aim of the present paper to give a different derivation to that in [17] of the Lebesgue measure decompositions implied by (3) and (5). This derivation, to be carried out in Section 2, is deduced from knowledge of a matrix change of variables formula core to the study of the so-called Wishart matrices in random matrix theory, which in turn is equivalent to a matrix polar integration theorem.…”

Section: Introductionmentioning

confidence: 96%

“…The matrix polar integration theorem can then, following [17], be used to derive Blashke-Petkantschin type decomposition of measure formulas, now for the real, complex and real quaternion cases. This is done in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Forrester¹

2020

MMJ

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In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S. R. Moghadasi has shown how the decomposition of the N-fold product of Lebesgue measure on R n implied by matrix polar decomposition can be used to derive the Blaschke-Petkantschin decomposition of measure formula from integral geometry. We use known formulas from random matrix theory to give a simplified derivation of the decomposition of Lebesgue product measure implied by matrix polar decomposition, applying too to the cases of complex and real quaternion entries, and we give corresponding generalisations of the Blaschke-Petkantschin formula. A number of applications to random matrix theory and integral geometry are given, including to the calculation of the moments of the volume content of the convex hull of k ≤ N + 1 points in R N , C N or H N with a Gaussian or uniform distribution.

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Section: Moments Of Determinantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Forrester¹

2020

MMJ

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“…The moments for the Gaussian weight for all three number systems, and similarly the moments for the beta type I weight with ν l = 0 (uniform on the sphere) and βν l /2 = 1 (uniform on the ball), have been computed in the recent work [12,Prop. 8]; see also [22] in the Gaussian case. The working relies on (Miles version of) the Blaschke-Petkantschin formula from integral geometry.…”

Section: Isotropic Rowsmentioning

confidence: 99%

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Forrester

Zhang

2020

J Stat Phys

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A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices X, the full Lyapunov spectrum for products of random matrices I N + 1 c X is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.

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“…After finishing the first version of the paper, the author became aware of close works by Moghadasi [19] and Forrester [10] devoted to application of the matrix polar decomposition to derivation of the Blaschke-Petkantschin formula. Our reasoning essentially differs from [10,19].…”

Section: Introductionmentioning

confidence: 99%

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

Rubin

2018

FCAA

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The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on R n in terms of the corresponding measures on k-dimensional linear subspaces of R n . We suggest a new elementary proof of this formula and discuss its connection with the celebrated Drury's identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. We give a new derivation of this identity and provide it with precise information about constant factors and the class of admissible functions.2010 Mathematics Subject Classification. Primary 44A12; Secondary 28A75, 60D05.

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POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝⁿ

Cited by 5 publications

References 4 publications

Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

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