|- | (conjugate transpose) || || |- | n is a positive integer || || |- | | | |- | | | |- | is diagonalizable f is differentiable at every eigenvalue | | |} In the last row, is the Kronecker delta and is the set of orthogonal projection operators that project onto the k-th eigenvector of X. Q is the matrix of eigenvectors of , and are the eigenvalues. The matrix function is defined in terms of the scalar function for diagonalizable matrices by where with . To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities: {|class="wikitable" style="text-align: center;" |+ Conversion from differential to derivative form ! Canonical differential form !! Equivalent derivative form (numerator layout) |- | || |- | || |- | || |- | || |- | || |- | || |} Applications Matrix differential calculus is used in statistics and econometrics, particularly for the statistical analysis of multivariate distributions, especially the multivariate normal distribution and other elliptical distributions. It is used in regression analysis to compute, for example, the ordinary least squares regression formula for the case of multiple explanatory variables. It is also used in local sensitivity and statistical diagnostics. See also Derivative (generalizations) Product integral Ricci calculus Notes References . Further reading . Note that this Wikipedia article has been nearly completely revised from the version criticized in this article. External links Software MatrixCalculus.org, a website for evaluating matrix calculus expressions symbolically NCAlgebra, an open-source Mathematica package that has some matrix calculus functionality SymPy supports symbolic matrix derivatives in its matrix expression module, as well as symbolic tensor derivatives in its array expression module. Information Matrix Reference Manual, Mike Brookes, Imperial College London. Matrix Differentiation (and some other stuff), Randal J. Barnes, Department of Civil Engineering, University of Minnesota. Notes on Matrix Calculus, Paul L. Fackler, North Carolina State University. Matrix Differential Calculus (slide presentation), Zhang Le, University of Edinburgh. Introduction to Vector and Matrix Differentiation (notes on matrix differentiation, in the context of Econometrics), Heino Bohn Nielsen. A note on differentiating matrices (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive. Vector/Matrix Calculus More notes on matrix differentiation. Matrix Identities (notes on matrix differentiation), Sam Roweis. Matrix theory Linear algebra Multivariable calculus