Some Applications of Matrix Derivatives in Multivariate Analysis

Abstract It is claimed that the reasons for using matrices of derivatives, in appropriate situations, are as compelling as those for using matrices. This paper provides basic material for such use. Different types of matrix derivatives are defined and illustrated. Simple and easy techniques are then derived and are shown to be applicable to a considerable collection of matrix functions. Applications are made to such problems as establishing matrix integrals from scalar ones, determining maximum likelihood estimates for complex likelihood functions, optimizing matrix functions when there are matrices of side conditions, and evaluating the Jacobians of certain classes of transformations. The emphasis is on simplicity of derivation and on breadth of application.

[1]  H. W. Turnbull.  On Differentiating a Matrix , 1928 .

[2]  H. W. Turnbull.  A Matrix Form of Taylor's Theorem , 1930 .

[3]  H. W. Turnbull.  Matrix Differentiation of the Characteristic Function , 1931 .

[4]  W. J. Duncan,et al.  Elementary matrices and some applications to dynamics and differential equations , 1939 .

[5]  A. D. Michal.  Matrix and tensor calculus , 1947 .

[6]  P. S. Dwyer,et al.  Symbolic Matrix Derivatives , 1948 .

[7]  Ingram Olkin,et al.  THE JACOBIANS OF CERTAIN MATRIX TRANSFORMATIONS USEFUL IN MULTIVARIATE ANALYSIS: BASED ON LECTURES OF P. L. HSU AT THE UNIVERSITY OF NORTH CAROLINA, 1947 , 1951 .

[8]  Ingram Olkin,et al.  NOTE ON ‘THE JACOBIANS OF CERTAIN MATRIX TRANSFORMATIONS USEFUL IN MULTIVARIATE ANALYSIS’ , 1953 .

[9]  Calyampudi R. Rao,et al.  Advanced Statistical Methods in Biometric Research. , 1953 .

[10]  P. S. Dwyer.  Generalizations of a Gaussian Theorem , 1958 .

[11]  S. Kullback,et al.  Some Aspects of Multivariate Analysis. , 1958 .

[12]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[13]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[14]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[15]  A. E. Maxwell,et al.  Factor Analysis as a Statistical Method. , 1964 .

[16]  G. Enderlein.  Wilks, S. S.: Mathematical Statistics. J. Wiley and Sons, New York–London 1962; 644 S., 98 s , 1964 .

[17]  R. Bargmann,et al.  MAXIMUM LIKELIHOOD ESTIMATION WITH INCOMPLETE MULTIVARIATE DATA , 1964 .

[18]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[19]  K. Jöreskog.  Testing a simple structure hypothesis in factor analysis , 1966, Psychometrika.

[20]  P. Schönemann,et al.  A generalized solution of the orthogonal procrustes problem , 1966 .