Semi-Inversion of a Diagonalizable Nonsingular Matrix

Donald R. Burleson, Ph.D.

Copyright © June 2017, April 2021 by Donald R. Burleson. All rights reserved.

In my previous article “The Semi-Inversion Operator and the Pauli Particle-Spin Matrices,” at www.blackmesapress.com/Semi-inverses.htm , I proposed the
concept of the (principal) semi-inverse of a nonsingular matrix A, so designated
because by the proposed transform φ(A) = A^{i} , a successive application of the
transform formally gives
the ordinary inverse of A.
This semi-inversion operator was described as essentially being handled
computationally by the eigenvalue relation
extended to
exponentiation of the eigenvalues
by the power
.

In the earlier article I developed a formula for semi-inverting a 2X2 nonsingular matrix with distinct eigenvalues, where the same general approach (more laboriously) can be pursued for larger matrices.

A considerable gain in ease of computation of the semi-inverse is at hand
when the square nonsingular matrix, of whatever size, is diagonalizable, whether
its eigenvalues are distinct or not (i.e. provided the vector space F_{n X 1} has an
eigenvector basis, a necessary and sufficient condition for diagonalizability). It
seems prudent at this point simply to state and prove the following:

THEOREM: Every diagonalizable nonsingular matrix is semi-invertible.

PROOF: Let A be any diagonalizable and invertible matrix. Then as A is
diagonalizable there is a nonsingular modal matrix M such that
, where Dg(λ_{j}) denotes the diagonal matrix having the
eigenvalues of A on the diagonal, and where M, as is customary, is formed by
taking respective eigenvectors as columns. But then exponentiation of A by semi-inversion merely consists of formally computing A^{i} as
Alternatively, since a matrix A is diagonalizable if and only if it has a spectral
decomposition
(where the matrices E_{j} are the principal
idempotents of A and the summation is taken over the distinct eigenvalues
comprising the spectrum of A), and since for any holomorphic function f the
matrix f(A) (with A diagonalizable) can be computed as
,
it follows that the semi-inverse of a diagonalizable nonsingular matrix A may be
computed as
, where the exponentiations on the eigenvalues
(nonzero since A is invertible) are defined by Euler’s formula
, i.e. since
we have, for
,
(λ_{j})^{ i} = cos ln (λ_{j}) + *i* sin ln (λ_{j}) , and likewise if
we
take
which in principal complex value as implied by
Euler’s relation
may be characterized as
,
. The result of this process is the
semi-inverse of A given itself already in diagonalized form, and in this form it is
evident that a second exponentiation of the same kind produces eigenvalues that
are the reciprocals of the original ones, in keeping with the fact that the inverse of
A has reciprocal eigenvalues. That is to say, the semi-inverse of the semi-inverse
is the inverse. This completes the proof.

COROLLARY: If a matrix A is diagonalizable and nonsingular and if z is any
complex number, then the matrix A^{z} exists.

PROOF: By the theorem, the semi-inverse of A is well-defined, and if z = x+yi one may compute by diagonalization.

EXAMPLE 1: Let A =

with eigenvalues
. Since A is 3X3 and has three
distinct (and nonzero) eigenvalues, A is diagonalizable (and nonsingular). For the
diagonalization of A, the modal matrix M will have as its columns the respective
eigenvectors (1,0,0)^{T}, (1,0,-1)^{T}, and (2,3,1)^{T} so that

The canonical diagonalization of A is then

and if φ denotes the matrix-valued transform that maps A to its semi-inverse, then the semi-inverse of A is found by simply replacing each eigenvalue in the diagonal matrix with its exponentiation, where in principal complex value The result would work out to be

Alternatively, from the diagonalization A = M[Dg(λ_{j})]M^{-1} we could have
determined the principal idempotent decomposition (spectral decomposition)

and the semi-inversion is performed by simply exponentiating the eigenvalues

1, -1, and 2 with the same results as before.

EXAMPLE 2: Even if the eigenvalues are not all distinct (so long as there is still an eigenbasis so that the matrix is diagonalizable) the semi-inverse can be computed the same way as in the previous example. E.g. for

with eigenvalues and eigenvectors

(1,2,1)^{T} , (-1,1,0)^{T} , and (-1,0,1)^{T} respectively, we have the modal matrix

with inverse

so that from the resulting diagonalization the desired semi-inverse is

= .

It should be mentioned that in terms of the techniques described here, a diagonalizable matrix A always needs also to be invertible (diagonalizability and invertibility being independent conditions) because otherwise if one of the eigenvalues were the corresponding eigenvalue exponentiations would have to include the undefined matrix element and if such an element were to occur then a further exponentiation would yield which of course is meaningless. But the combined requirements of invertibility and diagonalizability are always sufficient for the semi-inverse of a matrix to exist.

However, consider also the following.

EXAMPLE 3: Let The eigenvalues of A are

, and since one may verify that the only eigenvectors are those belonging to the eigenspace consisting of multiples of the vector so that there is no eigenbasis for the column space of A, the matrix A is not diagonalizable. However, there is a way to semi-invert A. If we examine powers of A:

etc., we observe an emerging pattern suggesting that for any positive integer n,

That this is indeed the case can easily be shown by mathematical induction: if k is any value for which the pattern holds, then

which is the matrix A to the power n for n = k+1. Now when we put

to formally produce , we may verify that this result is consistent with the definition of the principal semi-inverse of A by examining, as before, the powers of this semi-inverse itself:

etc. producing a pattern giving (as provable again by mathematical induction)

in which putting

produces which is in fact the inverse of A as needed. Thus A has worked out to be semi-invertible although not diagonalizable.

Altogether we have proved the following:

THEOREM: For nonsingular matrices, diagonalizability is a sufficient but not in general a necessary condition for semi-invertibility.