[fpr 2020] Fwd: Special seminar



maeda (at) ism.ac.jp         〒106-8569(個別番号)
Tel 03-5421-8734(直通) Fax 03-5421-8796(共用)

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> Date: Sun, 8 Jul 2001 02:13:49 -0400 
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> From: Yasumasa Baba <baba (at) ism.ac.jp> 
> Subject: Special seminar 
> 特別セミナーのご案内 
> 統計数理研究所 
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>   連絡先:baba (at) ism.ac.jp 
> ----------------------------------------------------- 
> <特別セミナーのお知らせ> 
> 日 時:平成13年7月24日(火) 15:00〜16:30 
> 場 所:統計数理研究所 2F研修室 
> 講演者:John Gower (Department of Statistics, The Open University, UK) 
> 演 題:Application of the Modified Leverier-Fedeev Algorithm 
>  講演者のJohn Gower
> 先生は、データ解析、計量生物統計等、様々な分野で顕著な業績を上げられていま
> す。この度は、大学入試センターの招聘と、大阪で開催される国際計量心理学会(
> IMPS-2001、7月15-19日)の招待講演ご発表のために来日されますので、東京滞在
> の機会を移用しましてこのセミナーを計画しました。 
>  なお、7月23日には大学入試センターでバイプロットについての講演をされるこ

> ニになっています。 
> ----------------------------------------------------- 
> IMPS-2001 
> <http://www.ir.rikkyo.ac.jp/imps2001/>http://www.ir.rikkyo.ac.jp/imps2001/ 
> --------------------------------------------- 
> -------------------------------------- 
> Yasumasa Baba 
> Professor 
> The Institute of Statistical Mathematics 
> 4-6-7 Minami-Azabu, Minato-ku 
> Tokyo 106-8569 
> Phone +81-3-5421-8739 
> Fax     +81-3-5421-8796 
> --------------------------------------

Applications of the Modified Leverrier-Faddeev 
John C. Gower
Department of Statistics, The Open University
Walton Hall, Milton Keynes, MK7 6AA, UK. 

The modified Leverrier Algorithm defines the sequence X_0 = A, X_I= 
AX_{I-1} + q_I A for i = 1,2,...,m, where m is the degree of the minimal 
polynomial of the square matrix A. It may be shown (i) that X_m= 0, (ii) a 
generalised inverse of A is easily obtained as a linear combination of 
X_{m-1} and X_{m-2}, (iii) the columns (rows) of X =sum_{i=0}^{m-1} 
\lambda^{m-i-1}X_i  give the post (pre) eigenvectors associated with any 
eigenvalue \lambda of A. Under mild conditions, rank(X) is the multiplicity 
of \lambda. The algorithm has little computational value but when A has a 
structured form that is preserved under summing and powering, then the form 
of X is known, without requiring any knowledge of the minimal polynomial, the 
terms X_I, or the eigenvalues of A. It is then often easy to deduce algebraic 
forms for the spectral decompositions, singular value decompositions and g-
inverses of A. 
I shall give examples of the many occasions where I have found the 
algorithm useful, some of which are given in the list of references. Two 
recent applications have been to multiple correspondence analysis of 
patterned Burt matrices and to the elucidation of the relationships between 
conventional dyadic distances and triadic distances. 
Explicit spectral decompositions, explicit g-inverses, explicit singular val
decompositions, statistics

Denis, J-B., & Gower, J.C. (1994) Asymptotic covariances for the parameters 
  biadditive models. Utilitas Mathematica, 46, 193-205. 
Denis, J-B., & Gower, J.C. (1996) Asymptotic confidence regions for biadditi
  models: Interpreting genotype-environment interactions. Applied Statistics, 
  45, 479-493. 
Gower, J.C. (1980) A modified Leverrier-Faddeev algorithm for matrices with 
  multiple eigenvalues. Linear Algebra and its Applications, 31, 61-70. 
Gower, J.C. (1980). An application of the Leverrier-Faddeev algorithm to 
  skew-symmetric matrix decompositions. Utilitas Mathematica 18, 225-240. 
Gower, J.C. & Groenen, P.J.F. (1991). Applications of the modified Leverrier-
   Faddeev algorithm for the construction of explicit matrix spectral 
   decompositions and inverses. Utilitas Mathematica 40, 51-64. 

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