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Functions of Matrices and Jacobians of Transformations

In: THE MULTIVARIATE NORMAL DISTRIBUTION Theory and Applications

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  • PHAM-GIA THU

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

Suggested Citation

  • Pham-Gia Thu, 2021. "Functions of Matrices and Jacobians of Transformations," World Scientific Book Chapters, in: THE MULTIVARIATE NORMAL DISTRIBUTION Theory and Applications, chapter 11, pages 347-376, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789811235290_0011
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    Keywords

    Univariate Normal; Multivariate; Random Variable; Random Vector; Random Matrix; Applications: Education; Psychology; Quality Control; Pattern Recognition;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics

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