Semi-Inversion of a Diagonalizable Nonsingular Matrix

Donald R. Burleson, Ph.D.

Copyright © June 2017 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 (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(_{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 ,
(_{j})^{ i} = cos ln (_{j}) + *i* sin ln (_{j}) , and likewise if we
take which in principal complex value as implied by
Euler's relation may be characterized as
, .
In the case of a non-real complex eigenvalue z, the necessary exponentiation in that term of the spectral decomposition
can be computed by evaluating ln(z) in the expressions cos ln(z) and sin ln(z)
as ln(z) = ln(/z/) + iArg(z), in principal complex value. 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: **Every nonsingular Hermitian matrix is semi-invertible.

**PROOF:** Every Hermitian matrix is diagonalizable.

**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 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(_{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.

Let with eigenvalues and with inverse

. A is invertible but not diagonalizable because its double eigenvalue generates an eigenspace that is only one-dimensional, i.e. there is no eigenbasis. But we may still semi-invert this matrix.

In general I define a semi-inversion operator as a matrix-valued transform that has, for matrices in its domain, the property . Then consider the motivation provided by the fact that one may easily show by mathematical induction that for n any positive integer (reflecting powers of A) or any negative integer (reflecting powers of its inverse). So for any matrix of the form B = it would be natural to define (B) = with the result

as required. For invertible but non-diagonalizable matrices of the form

one may similarly define the semi-inverse as .

Thus by example together with the results obtained in the previous article cited, we have proved the following:

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