The Spectral Radii of Semi-inverses of Diagonalizable Matrices
Donald R. Burleson, Ph.D.
Copyright (c) 2019 by Donald R. Burleson. All rights reserved.
In an earlier research article I have proven a theorem to the effect that every diagonalizable nonsingular matrix is semi-invertible and has a countably
infinite set of semi-inverses forming a matrix sequence that converges to the null matrix. We now pursue an implication of this result.
The proof of the stated theorem has depended on the fact that a matrix A is diagonalizable if and only if it has a spectral decomposition (principal idempotent decomposition)
A = Σj λjEj
where the coefficients λj are the eigenvalues of A, and we consider the following details.
For the semi-inversion operator φ(A) = Ai, or rather (since Ai is multi-valued) the sequence of semi-inversion operators φn(A) for n = 0, 1, 2, ...,
the function f(x) = xi being holomorphic (in all its branches) in any open neighorhood in the complex plane not containing the origin, we may take
φn(A) = Σj(λj)iEj
where by Euler's formula the multi-valued eigenvalues (λj)i of Ai are given by
λji = 1i(cos ln λj + i sin ln λj)
= e-2nπ(cos ln λj + i sin ln λj).
(A coefficient of e2nπ produces the inverses of these semi-inverses.)
One notes that in the complex plane the absolute value of the λji is uniformly
e-2nπ[cos2(ln λj) + sin2(ln λj)]1/2 = e-2nπ,
which of course for n = 0 is unity. This serves to prove the following
THEOREM. For each semi-inverse φn(A) of a diagonalizable nonsingular matrix A,
the eigenvalues of φn(A) in the complex plane have uniform absolute value and lie on
a circle of radius e-2nπ centered at the origin.
REMARKS: Since lim e-2nπ = 0, as one progresses through the semi-inverses φn(A) the spectral radii of these matrices form a sequence converging to 0.
Geometrically, in the complex plane the circles containing the eigenvalues (starting with the unit circle for the case n = 0) "collapse" toward a single point (the origin)
in the limit, exponentially fast, having radii 1, e-2π, e-4π, e-6π, etc. (On the sixth pass, the spectral radius is already
less than 10-6.) In a sense, this predictable
nature of the eigenvalues makes the semi-inverse matrices typically "better behaved" than the original matrix itself.
The semi-inverses in effect are islands of order situated transformationally "between" an invertible matrix and its inverse.