Singular random matrix decompositions: Jacobians

Abstract

For a singular random matrix X ; we find the Jacobians associated to the following decompositions: QR; Polar, Singular Value ðSVDÞ; L 0 U; L 0 DM and modified QR (QDR). Similarly, for the cross-product matrix S ¼ X 0 X we find the Jacobians of the Spectral, Cholesky's, L 0 DL and symmetric nonnegative definite square root decompositions. r 2004 Elsevier Inc. All rights reserved.

Singular extended skew-elliptical distributions

“…If q = min(r, k), explicit expressions for (dY) can be given as functions of the QR, Polar, Singular value and QR modified decompositions, see Díaz-García and González-Farías (2005a). Now, Khatri (1968) shows that the density function (2) is not unique, see also Díaz-García and González-Farías (2005b); however, it is important to note that once a distribution expression (2) is found, the results do not depend on the selected density, see Rao (1973).…”

Principal components under singularity

“…and m respectively, but with respect to the measures of Hausdorff (dS) and (dX) defined in Díaz-García and González-Farías (2005a). Now, let l 1 , .…”

“…). Nevertheless, if there is a linear dependence between the columns, for example r < m are linearly independent, then X now has density with respect to the Hausdorff measure, see or Díaz-García and González-Farías (2005a). Still more, if it is assumed that the sample comes from a population with m−dimensional normal ditribución,…”

“…if X has singular matrix normal distribution (see and Khatri (1968)), whose density exists with respect to the Hausdorff measure; then, in such case, as much the density as the measure are not unique and their explicit forms will depend on the base and the set of coordinates selected for the subspace in where such density exists, see Díaz-García and González-Farías (2005a). An analogous situation appears with the distribution of the sample covariance matrix S, in that case nS has a m− dimensional singular Wishart distribution of rank r, nS ∼ W r m ((N − 1), Σ) whose density also exists with respect to the Hausdorff measure defined in the corresponding subspace, see , and Díaz-García and González-Farías (2005b).…”

Some generalisations of Birnbaum-Saunders and sinh-normal distributions

“…In particular, these results are applicable to the case of a random vector. But, if X ∈ m , the SVD of the vector X is given simply by X = ρW, where W ∈ m is W 2 = 1 and ρ = X , and here the vectorial cases for the SVD, the QR decomposition and the polar decomposition coincide, see [7]. Moreover, the decomposition X = ρW, for a random vector with spherical (or elliptical) distribution, is known in the literature as the stochastic representation of the vector X, see [ …”