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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: In this paper, the expectation and covariance matrix of the Wishart distribution are derived, where the expectation is derived from the expectation matrix of a square matrix containing only zeros and ones.
Abstract: The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.
469 citations
Book•
01 Jan 2008
TL;DR: The most common matrix function is the matrix inverse, which is not treated specifically in this chapter, but is covered in Section~1.5 and Section~51.3 as discussed by the authors.
Abstract: Matrix functions are used in many areas of linear algebra and arise
in numerous applications in science and engineering.
The most common matrix function is the matrix inverse;
it is not treated specifically in this chapter,
but is covered in Section~1.5 and Section~51.3.
This chapter is concerned with general matrix functions as
well as specific cases such as matrix square roots,
trigonometric functions,
and the exponential and logarithmic functions.
455 citations
TL;DR: The aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
Abstract: Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
451 citations
TL;DR: This survey paper describes how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what the authors call displacement structure, and develops a fast triangularization procedure.
Abstract: In this survey paper, we describe how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what we call displacement structure. In particular, a fast triangularization procedure can be developed for such matrices, generalizing in a striking way an algorithm presented by Schur (1917) [J. Reine Angew. Math., 147 (1917), pp. 205–232] in a paper on checking when a power series is bounded in the unit disc. This factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra. We mention, among others, inverse scattering, analytic and unconstrained rational interpolation theory, digital filter design, adaptive filtering, and state-space least-squares estimation.
447 citations
Posted Content•
TL;DR: In this article, the expectation and covariance matrix of the Wishart distribution are derived, where the expectation is derived from the expectation matrix of a square matrix containing only zeros and ones.
Abstract: The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.
443 citations