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 = Σ_{j} λ_{j}E_{j}

where the coefficients λ_{j} are the eigenvalues of A, and we consider the following details.

For the semi-inversion operator φ(A) = A^{i}, or rather (since A^{i} is multi-valued) the sequence of semi-inversion operators φ_{n}(A) for n = 0, 1, 2, ...,

the function f(x) = x^{i} being holomorphic (in all its branches) in any open neighorhood in the complex plane not containing the origin, we may take

φ_{n}(A) = Σ_{j}(λ_{j})^{i}E_{j}

where by Euler's formula the multi-valued eigenvalues (λ_{j})^{i} of A^{i} are given by
λ_{j}^{i} = 1^{i}(cos ln λ_{j} + i sin ln λ_{j})

= e^{-2nπ}(cos ln λ_{j} + i sin ln λ_{j}).

(A coefficient of e^{2nπ} produces the inverses of these semi-inverses.)

One notes that in the complex plane the absolute value of the λ_{j}^{i} is uniformly

e^{-2nπ}[cos^{2}(ln λ_{j}) + sin^{2}(ln λ_{j})]^{1/2} = e^{-2nπ},

which of course for n = 0 is unity. This serves to prove the following

**THEOREM.** For each semi-inverse φ_{n}(A) of a diagonalizable nonsingular matrix A,

the eigenvalues of φ_{n}(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 φ_{n}(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.