Functions of Matrices and Jacobians of Transformations

Pham-Gia Thu

Chapter 11 in The Multivariate Normal Distribution:Theory and Applications, 2021, pp 347-376 from World Scientific Publishing Co. Pte. Ltd.

Abstract: The following sections are included:IntroductionReview on MatricesLinear transformations and matricesBasic propertiesImportant decompositions of a matrixTwo Important Operations on MatricesJacobians of Transformations Between Special MatricesMatrices of functions and functions of matricesTransformations in RpJacobians in changing variablesJacobian and exterior productThe multigamma functionDerivatives of Vectors and MatricesDerivatives of a vectorDerivatives of a matrix and of a function of a matrixJacobian for the inverse matrixPowers of a square matrixDifferentials in Matrix DecompositionsJacobians Associated with EigenvaluesIntegration of a MatrixUse of skew-symmetric matrices (Xt = −X)Decomposition with one component skew-symmetricSpectral decompositionSingular value decomposition of a square matrixComputation of a Simple Definite Matrix Integral Under Various HypothesesStiefel Manifold and the Orthogonal GroupConclusionBibliography

Keywords: Univariate Normal; Multivariate; Random Variable; Random Vector; Random Matrix; Applications: Education; Psychology; Quality Control; Pattern Recognition (search for similar items in EconPapers)
JEL-codes: C01 C1 C15 C4 C46 (search for similar items in EconPapers)
Date: 2021
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