Properties of the matrix A-XY

AbstractThe main topic of this paper is the matrix V=A−XY*, where A is a nonsingular complex k×k matrix and X and Y are k×p complex matrices of full column rank. Because properties of the matrix V can be derived from those of the matrix Q=I−XY*, we will consider in particular the case where A=I. For the case that Y*X=I, so that Q is singular, we will derive the Moore–Penrose inverse of Q. The Moore–Penrose inverse of V in case Y*A−1X=I then easily follows. Finally, we will focus on the eigenvalues and eigenvectors of the real matrix D−xy′ with D diagonal

Contributions to multivariate analysis with applications in marketing

Dit proefschrift behandelt een aantal onderwerpen uit de multivariate analyse, waarbij het begrip ‘multivariate analyse’ ruim moet worden ge¨ınterpreteerd. Naast onderwerpen uit de multivariate statistiek in enge zin, besteden we ook aandacht aan matrixrekening, ‘sum-constrained linear models’, marketing en tekstanalyse.

Properties of the matrix A-XY

The main topic of this paper is the matrix V = A - XY*, where A is a nonsingular complex k x k matrix and X and Y are k x p complex matrices of full column rank. Because properties of the matrix V can be derived from those of the matrix Q = I - XY*, we will consider in particular the case where A = I. For the case that Y* X = I, so that Q is singular, we will derive the Moore-Penrose inverse of Q. The Moore-Penrose inverse of V in case Y*A(-1)X = I then easily follows. Finally, we will focus on the eigenvalues and eigenvectors of the real matrix D - xy' with D diagonal. (c) 2004 Elsevier Inc. All rights reserved

Properties of the matrix A- XY

As a management problem the identification of stakeholders is not easily solved. It comprises a modelling and a normative issue, which need to be solved in connection with each other. In stakeholder literature knowledge can be found, e.g. on various stakeholder categorizations, that could be useful for the modelling issue. However, the normative issue remains unresolved. Furthermore, the modelling of the so-called stakeholder category “the affected” is even more difficult. Nevertheless, this group holds justified interests in aspects of organizational activity and are, for that reason, legitimate stakeholders. In this article it is explored to what extent Critical Systems Heuristics can help resolving the managerial problem of identifying stakeholders, particularly the affected. Critical Systems Heuristics can be viewed a modelling methodology. The normative aspect of modelling is crucial in this methodology. Using the distinction between “the involved” and “the affected” a variety of boundary judgments is discussed. Special attention is given to the so-called “witness” as a representative of the affected.

The rank of a normally distributed matrix and positive definiteness of a noncentral Wishart distributed matrix

If , then S=X'X has the noncentral Wishart distribution , where [Lambda]=M'M. Here [Sigma] is allowed to be singular. It is well known that if [Lambda]=0, then S has a (central) Wishart distribution and S is positive definite with probability 1 if and only if n[greater-or-equal, slanted]k and [Sigma] is positive definite. We show that if S has a noncentral Wishart distribution, then S is positive definite with probability 1 if and only if n[greater-or-equal, slanted]k and [Sigma]+[Lambda] is positive definite. This is a consequence of the main result that with probability 1.Multivariate normal distribution Singular covariance matrix Rank Noncentral Wishart Positive definite

The rank of a normally distributed matrix and positive definiteness of a noncentral Wishart distributed matrix

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