Well, Mr. Jacobi, here it is: all the generalized inversion of two generations of invertors who, knowingly or unknowingly, subscribed to (and extended) your dictum. Please forgive us if we have over-inverted, or if we have not always inverted in the natural and sensible way. (Some of us have inverted with labor and pain by using hints from a dean or a tenure and promotion committee that “you better invert more, or else you would be inverted.”)

Annotated Bibliography on Generalized Inverses and Applications

We say that a unitary matrix R is rotative (specifically, k-rotative) if its minimal polynomial is xk 1 for some k 2. Let R 2 Cm m and S 2 Cn n be k-rotative, , ˇ, 2 f0; 1; : : : ; k 1g, and ˇ ¤ 0. Let D e i=k . We define A.R; S; ; ˇ; / to be the class of matrices A 2 Cm n such thatR ASˇ D A. If m D n and S D R, we denote the class by A.R; ; ˇ; /. We characterize the class A.R; S; ; ˇ; / and discuss the problem of Moore–Penrose inversion of a wider class of matrices that includes A.R; S; ; ˇ; /. Under the additional assumption that. ; k/ D .ˇ; k/ D 1, we give a representation of a matrix A in A.R; S; ; ˇ; / in terms of matrices Fs 2 Ccs ds , where Pk 1 sD0 cs D m and Pk 1 sD0 ds D n, and show that Moore–Penrose inversion, singular value decomposition, and the least squares problem for such a matrix reduce respectively to the same problems for F0, . . . , Fk 1. We consider the eigenvalue problem for matrices inA.R; ; ˇ; /. We study a class of generalized circulants generated by blocks A0, . . . , Ak 1 2 Cd1 d2 , and show that they are in A.R; S; 1; ˇ; / for suitable choices of R, S , and . In this case we give explicit formulas for F0 , . . . , Fk 1 in terms of A0 , . . . , Ak 1, and for A  in terms of F  0 , . . . , F  k 1 .

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On matrices with rotative symmetries

When $(n,r) = 1$, the Moore–Penrose pseudo-inverse of an $n \times n$r-circulant matrix is a strong spectral inverse.

W. T. Stallings

Papers

A Strong Spectral Inverse for an r-Circulant Matrix