文档介绍：Notes on Matrix Calculus Paul L. Fackler ? North Carolina State University September 27, 2005 Matrix calculus is concerned with rules for operating on functions of matrices. For example, suppose that anm×nmatrixXis mapped into a p×qmatrixY. We are interested in obtaining expressions for derivatives such as?Y ij ?X kl , for alli, jandk, l. The main di?culty here is keeping track of where things are put. There is no reason to use subscripts; it is far better instead to use a system for ordering the results using matrix operations. Matrix calculus makes heavy use of the vec operator and Kronecker products. The vec operator vectorizes a matrix by stacking its columns (it is convention that column rather than row stacking is used). For example, vectorizing the matrix ??? 1 2 3 4 5 6 ????Paul L. Fackler is an Associate Professor in the Department of Agricultural and Re- source Economics at North Carolina State University. These notes are copyrighted mate- rial. They may be freely copied for individual use but should be appropriated referenced in published work. Mail: Department of Agricultural and Resource Economics NCSU, Box 8109 Raleigh NC, 27695, USA e-mail:******@ Web-site:/～pfackler/ c°2005, Paul L. Fackler 1 produces ????????? 135246 ????????? The Kronecker product of two matrices,AandB, whereAism×nandB isp×q, is de?ned as A?B= ????? A 11B A 12B . . . A 1nB A 21B A 22B . . . A 2nB . . . . . . . . . . . . A m1B A m2B . . . A mnB ?????, which is anmp×nqmatrix. There is an important relationship between the Kronecker product and the vec operator: vec(AXB) = (B >?A)vec(X). This relationship is extremely useful in deriving matrix calculus results. Another matrix operator that will prove useful is one related to the vec operator. De?ne the matrixT m,nas the matrix that transformsvec(A) into vec(A >): T m,nvec(A) = vec(A >). Note the size of this matrix ismn× m,nhas a number of special properties. The ?rst is clear from