Yoramar Magnus , Neudecker : The Commutation Matrix: Some Properties and Applications It fills a long-felt need for an exhaustive, unified and self-contained treatment of matrix theory and matrix differential calculus. MagnusHeinz Neudecker. You do not have access to this content. About the Author Jan R. Permanent link to this document https: Singular Wishart and multivariate beta distributions Srivastava, M.
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This chapter introduces the basics, which will be used throughout the text. For more information, I recommend four sources in particular. The most complete treatment, but not the easiest starting point, is the book by Magnus and Neudecker Download chapter PDF 2. The answer to my question was yes. It can be that simple. This book relies on this set of mathematical techniques. More accessible introductions can be found in the paper by Magnus and Neudecker and especially the text by Abadir and Magnus A review paper by Nel is helpful in placing the Magnus-Neudecker formulation in the context of other attempts at a calculus of matrices.
Sensitivity analysis asks how much change in an outcome variable y is caused by a change in some parameter x. At its most basic level, and with some reasonable assumptions about the continuity and differentiability of the functional relationships involved, the solution is given by differential calculus. However, the outcomes of a demographic calculation may be scalar-valued e.
Any of these outcomes may be functions of scalar-valued parameters e. Thus, sensitivity analysis in demography requires more than the simple derivative in 2.
The i, j entry of the matrix A is aij, and the ith entry of the vector n is ni. The transpose of X is XT. Logarithms are natural logarithms. The symbol 1 denotes a vector of ones. The vector ei is a unit vector with 1 in the ith entry and zeros elsewhere. The identity matrix is I. Where necessary for clarity, the dimension of matrices or vectors will be indicated by a subscript. In some places Chaps. The Kronecker product is defined for any sizes of A and B.
The vec operator and the vec-permutation matrix are particularly important in multistate models e. Extensions to an arbitrary number of dimensions, so-called hyperstate models, have been presented by Roth and Caswell What, however, does it mean to speak of the derivative of a scalar with respect to a vector, or of a vector with respect to another vector, or any other combination?
These can be defined in more than one way and the choice is critical Nel ; Magnus and Neudecker This book relies on the notation due to Magnus and Neudecker, because it makes certain operations possible and consistent. Derivatives involving matrices are written by first transforming the matrices into vectors using the vec operator, and then applying the rules for vector differentiation to the resulting vectors.
The Magnus-Neudecker notation, unlike some alternatives, extends the familiar scalar chain rule to derivatives of vectors and matrices Nel ; Magnus and Neudecker Checking dimensional consistency in this way is a useful way to find errors. Experience suggests that, for many readers of this book, this relationship is shrouded in the mists of long-ago calculus classes. This is customarily denoted dy, but for the moment, I will denote it by cy. Now, return to the standard notation of dy for the differential of y.
Mathematicians are strangely unconcerned with this ambiguity e. All this leads to a set of familiar rules for calculating differentials that guarantee that they can be used to create derivatives. For example, dividing 2. From 2. The differentials of an operators applied elementwise to a vector can be obtained from the differentials of the elements.
But, one cannot divide by a vector. This applies equally to more complicated expressions for the differential. Many more can be found in Magnus and Neudecker and Abadir and Magnus Diagonal matrices. Matrices other than I can be used in this way to mask entries of a matrix. Some Markov chain calculations Chaps. The Kronecker product. Differentiating the Kronecker product is a bit more complicated Magnus and Neudecker , Theorem This requires a result for the vec of the Kronecker product.
The matrix inverse. The square root and ratios. In calculating standard deviations and coefficients of variation it is useful to calculate the elementwise square root and the elementwise ratio of two vectors. The books by Magnus and Neudecker and Abadir and Magnus contain many other results, and demographically relevant derivations appear throughout this book, especially in Chap.
ISBN 13: 9780471915164
About this title This book provides a unified treatment of matrix differential calculus, specifically written for econometricians and statisticians. Divided into six parts, the book begins with a treatment of matrix algebra, discussing the Schur, Jordan, and singular-value decompositions, the Hadamard and Kronecker products, and more. The second section is the theoretical core of the book and presents a thorough development of the theory of differentials. Practically-oriented, part three contains the rules for working with differentials and lists the differentials of important scalar, vector, and matrix functions.
Matrix Calculus and Notation
Google Scholar Project Euclid. The Linear Regression Model. This is the newest edition covering information and developments since Neudecker Search this author in: Some Properties and Applications. Texts and References Section. First Order Differentials and Jacobian Matrices.
Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition