The Distributions of Latent Roots and Vectors

1Introduction Many problems in the multivariate analysis are concerned with the fbllowing diagonal transformations of matrices ALetbe a pp symetric matrix Then there exists an orthogonal trans- f()rmation matrix L such that L1)zLt whereDis a diagonal matrix whose diagonal elements are1p BLetandbe both symmetric pmatrices and be positive definite Then there exists a non singular matrix T such that u-TT v- TD) TX CLet Uand Vbe resptively pp andqqmatrices and Wapq rectangular matrix Then there exist non-singular matrices S and T such that v-T7 w- SDz(9) T where 1)is a pqrectangular diagonal matrix such as D(z69P) For instance the transformation A is used in the Principal component allalysis the transfbrmation B in the discriminant analysis and the multivariate analysis of variance and C in the callonical correlation analysis To deal with the transfbrmation C the fbllowg theorem is usefu1 DGiveamatrix(pxq) there exists a pair of orthogonal matrices L(pp) and4((7q)such that LD(P4)M Among the above fbur transf()rmations A alld D might be basic in the sense that the others can be constructed of the transfbrmations A and D In the next stion therefbre we shall start to give Jacobialls of the transformations A and D These transfbrmations have various interestillg properties especially extremal properties and uniqueness of transormation matrices The extremal property of transfbrmation A will be shown as fbllows A1 Letbe a symmetric and semi-positive definite pmatrix anda family ofmatrices C(s)such that a norm of every row vtor cof C is equal to 1(C(Cl Cs) CiCi 1) Then YTUMURA s (11) Max l CAC11 c8 i1 where 2 s are the latent roots ofarranged in the descending order and this maximum value can be attained by the matrix which consists of s latent vectors corresponding to Rs This extremal property is obtained if(11)is replaced by Max tr CAC i1 where C is an orthonormalP matrix About the transfbrmation matrix in A the uniqueness is well known The same conclusion can be obtained fbr all other transfbrmations For example D2 Letbe a pq matrix Suppose for convenience pq If all Iatent roots ofarenon-zero and different from each other and the Rs are arranged in the descending order in their absolute values then the transfbrmatimatrix L and only therst p rows of M are uniquely determined neglting their signsFrom this theorem the last q-p rows of M are completely arbitrary as long asent elements Similarly if q the matrix T in the transformation C is notiquely determined and the same thing fbr in the transfbrmation B if the rank of V is less than In these cases Jacobians of the transformations consequently the distributions of latent vtors do not exist in the usual sense Many authors have investigation on the distributions of latent roots Z in the transformations A-C In the central case the problem has been solved by Fisher9 Girshick10 and Hsu12