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Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics
Author(s) -
Henderson Harold V.,
Searle S. R.
Publication year - 1979
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315017
Subject(s) - jacobian matrix and determinant , diagonal , kronecker product , matrix (chemical analysis) , multivariate statistics , statistics , transformation (genetics) , column (typography) , mathematics , square matrix , product (mathematics) , kronecker delta , square (algebra) , row and column spaces , combinatorics , row , computer science , physics , chromatography , chemistry , symmetric matrix , eigenvalues and eigenvectors , geometry , biochemistry , quantum mechanics , connection (principal bundle) , database , gene
Abstract The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one‐to‐one transformation X → Y is then ∣∣∂(vecX)/∂(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ∂(vechX)/∂(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.