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_jE_j

where the coefficients λ_j 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 φ_n(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

φ_n(A) = Σ_j(λ_j)ⁱE_j

where by Euler's formula the multi-valued eigenvalues (λ_j)ⁱ of Aⁱ are given by λ_jⁱ = 1ⁱ(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.)

However, since the most general form of an eigenvalue λ is a (for our purposes nonzero) complex number a + bi, the logarithms in
the above expression must be taken to be (in principal value at any rate) complex numbers of the form ln|λ| + i Arg(λ).
Thus
λⁱ = e^-2nπcos[ln|λ| + i Arg(λ)] + i sin[ln|λ|) + i Arg(λ)]

which in the principal semi-inverse case with n = 0 (i.e. omitting for the moment the coefficient e^-2nπ) is:

cos(ln|λ|)cos[i Arg(λ)] - sin(ln|λ|)sin[i Arg(λ)] + i [sin(ln|λ|)cos(i Arg λ) + cos(ln|λ|)sin(i Arg λ)],

and that, using the facts that cos(i Arg λ) = (1/2)(e^{- Arg λ} + e^{Arg λ} and sin(i Arg λ) = (1/2i)(e^{- Arg λ}- e^{Arg λ}),
simplifies to

λⁱ = (cos ln|λ| + i sin ln|λ|)e^{- Arg(λ)}
in general having a coefficient of e^-2nπ for cases other than n = 0. This gives:

THEOREM: For any complex eigenvalue λ of a diagonalizable nonsingular matrix A, the corresponding eigenvalue of the principal semi-inverse
of A is λⁱ = |λ|ⁱe^{- Arg(λ)}, where |λ|ⁱ = cos ln|λ| + i sin ln|λ|. For the multivalued semi-inverses of A
the form of the eigenvalue(s) would be λⁱ = e^-2nπ|λ|ⁱe<sup>- Arg(λ).

REMARKS: Because of the coefficient of e-2nπ, as one progresses through the successive semi-inverses of A with n = 0,1,2,..., the spectral radius of A_ni approaches 0.

As special cases, if λ is real and positive, Arg(λ) = 0 and λi = cos ln(λ) + i sin ln(λ), and if λ is real and negative, Arg(λ) = π and λi = |λ|i e- π. Otherwise, for nonreal complex numbers one obtains such values as (e.g. for λ = 1 + i)
λi = |1 + i|ie- Arg(1 + i) = (1/2)(cos ln 2 + i sin ln 2)e- (1/4)π.</sup>