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.)

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

λ^{i} = e^{-2nπ}cos[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

λ^{i} = (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 λ^{i} = |λ|^{i}e^{- 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 λ^{i} = e^{-2nπ}|λ|^{i}e^{- 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 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.
}