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Communications of the ACM

Communications of the ACM

Pei matrix eigenvectors

M. L. Pei [Comm. ACM 5, 10 (Oct. 1962)] gave an explicit inverse for a matrix of the form M + &dgr;I, where M is an n-square matrix of ones and &dgr; is a nonzero parameter. The eigenvalues of the Pei matrix were given by W. S. LaSor [Comm. ACM 6, 3 (Mar. 1963)]. The eigenvectors may be obtained by considering the system (M+&dgrI)x = &lgr;x, the jth equation of which is S + &dgr;xj = &lgr;xj , (1) where S denotes ∑ni=1 xi. On summing the equations for j = 1, 2, ··· , n, we obtain nS + &dgr;S = &lgr;S. From this we conclude that (a) S = 0 or (b) &lgr; = n + &dgr;.

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