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.)
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).
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 + eArg λ and sin(i Arg λ) = (1/2i)(e- Arg λ- eArg λ),
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- Arg(λ).
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 Ani 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.