THE SEMI-INVERSION OPERATOR

AND THE PAULI PARTICLE-SPIN MATRICES

by

Donald R. Burleson, Ph.D.

Copyright © 2006 by Donald R. Burleson. All rights reserved.

To go to a related article, which has a link

to the “About the Author” page, click here.

For a square
matrix A with distinct nonzero eigenvalues λ_{j} and associated eigenvectors V_{j}, one may extend the relation

A^{x}V_{j}
= λ_{j}^{x}V_{j}

to complex values of x, and in particular to x = i, to define (in terms of principal complex value) the
matrix power A^{i}. For the
necessary eigenvalue powers λ_{j}^{i}
one may make use of Euler’s identity e^{iθ}
= cos θ + i sin θ
and its implied relation e^{i}^{(π +
2kπ)} = -1 to determine that in principal value (i.e. with k = 0) 1^{i}
= 1 and (-1)^{i}^{ }= e^{-π},
so that more generally we have not only a^{i}
= (e^{ln}^{(a)})^{i}^{
}= e^{i}^{ ln(a)}
= cos ln(a) + i sin ln(a) for a > 0 but
also, for a < 0,

say a = -b (with b > 0), a^{i}
= (-b)^{i}^{ }= (-1)^{i}b^{i} = e^{-π}[cos ln(b) + i
sin ln (b)].
In any case we can express each λ_{j}^{i}
(even when λ_{j} is nonreal,
it turns out) as a specific complex-number (in some cases real-number) value,
and this enables one to compute the elements of the matrix A^{i}.

The curious
thing about A^{i} is that since (A^{i})^{i}^{
}= A^{-1} (the ordinary inverse of A) one may designate the
operator Φ (on the full linear group) that maps A to Φ(A) = A^{i}
as the (principal) *semi-inversion
operator*, and may designate the image matrix A^{i} as the
(principal) *semi-inverse* of A. Since applying this operator twice in a row
takes A to A^{-1}, the semi-inverse A^{i} is, in a sense, a
matrix “halfway between” a nonsingular matrix A and its inverse. (In the case of self-inverse matrices like
those of Pauli, the semi-inverse
is “halfway between” a matrix and itself!)

It is
understood, when we refer to “the” semi-inverse, that the reference is to
principal complex value in all instances.
The matrix A^{-i }fulfils the same
role as A^{i}, and both A^{i} and A^{-i
}are multivalued due to the multivalued
nature of the eigenvalue exponentiations λ_{j}^{i}; but the term “semi-inverse”
herein assumes principal-valued entities throughout and refers to the
principal-valued matrix A^{i} as defined.

As a
special case, for a one-by-one matrix A = (a) regarded as isomorphic to a
scalar, for a > 0 we may observe that since matrix inversion corresponds in
this case to reciprocation, the result of applying the operator Φ may be
called the (principal) “semi-reciprocal” of the number a. For example, the semi-reciprocal of a = 2 is
the complex number a^{i} = cos ln(2) + i
sin ln(2) which is approximately 0.7692389 +
0.63896128i.

The
semi-inverse may generally be computed as follows for a nonsingular matrix with
distinct eigenvalues λ_{1} and λ_{2};
the illustration is for a 2X2 matrix

_{}

where we will initially assume b ≠ 0 and c ≠ 0. (If either b = 0 or c = 0 the semi-inverse will be determined as a special case.)

If V_{1}
= (x,y)^{T} is an eigenvector belonging to
the eigenspace corresponding to λ_{1},
we have

_{} whence ax + by =
λ_{1}x

from which we may infer an eigenvector of the form

_{}.

Likewise from

_{} whence cx + dy = λ_{2}y

we may infer an eigenvector of the form

_{}_{}

since trace(A) = a + d = λ_{1} + λ_{2}. Now let

_{}

Then from the relation A^{i}V_{j}
= λ_{j}^{i}V_{j} we have
(for V_{1})

_{}

Likewise (for V_{2}) we have

_{}

These matrix products imply

bP
+ (λ_{1}-a)Q = λ_{1}^{i}b

-(λ_{1}-a)P
+ cQ = -λ_{2}^{i}(λ_{1}-a)

and

bR
+ (λ_{1}-a)S = λ_{1}^{i}(λ_{1}-a)

-(λ_{1}-a)R
+ cS = λ_{2}^{i}c.

Solving for P, Q, R, and S gives, for

_{}

the results

_{}, _{}

_{} _{}

so that

_{}_{}

which may be rewritten

_{}_{}

It should be noted that while the symbol d does not appear here,
its involvement is implied, since for λ_{1}-a, we may always write
d – λ_{2}, and since the eigenvalues
λ_{1} and λ_{2} are functions of a, b, c, and d.

Here we assumed b ≠ 0 and c ≠ 0. If b = 0, similar computations show that

_{}_{}

since for this matrix we may take a = λ_{1} and
d = λ_{2}, and since the difference of the eigenvalues
is by hypothesis nonzero.

Likewise, if c = 0, we may compute

_{}.

It follows that if both b = 0 and c = 0,

_{}_{}

Clearly,
then, the semi-inverse matrix Φ(A) = A^{i} is well-defined for
matrices of this size, and similar considerations are feasible for larger
sizes.

We consider now an important subclass of 2X2 matrices.

Let U be
the set of all 2X2 matrices M having eigenvalues
λ_{1} = 1 and λ_{2} = -1, hence having trace(M) = 0
and det(M) = -1.
(This set U contains the well-known Pauli
matrices; more about this, in what follows.)

The following theorem shows that for these unit-eigenvalued matrices M, the semi-inverse can be computed easily as a simple (in fact, first-degree) polynomial function of M.

**THEOREM:** If M belongs to the class U, i.e. if M is a
2X2 matrix with trace(M) = 0 and det(M) = -1, then
the (principal) semi-inverse can be written

M^{i} = ½[I +
M + e^{-π}(I-M)],

where I is the 2X2 identity matrix. Equivalently, if one defines the polynomial function

g(x) = ½(1-e^{-π})x
+ ½(1+e^{-π}),

then the semi-inverse of M is given by

M^{i} = g(M).

**PROOF:**
Let

_{}

be any 2X2 matrix having trace(M) = 0 and det(M) = -1. The latter condition implies

a^{2} + bc = 1

so that

_{}

for b ≠ 0. (For the case b = 0 the theorem is readily verifiable as a special case.) So

_{}

Substituting into the general form obtained above, we have, for a matrix of this form:

_{}

_{}

_{}

One may then directly compute ½[I + M + e^{-π}(I-M)]
for comparison:

_{}

as already determined, which one can see by comparing matrices
element by element. Rewriting the
expression ½[I + M + e^{-π}(I – M)] as ½(1 – e^{-π})M
+ ½(1 + e^{-π})I shows that for this class of matrices the
generating polynomial is

g(x) = ½(1 – e^{-π})x
+ ½(1 + e^{-π}).
█

The
polynomial form given in the theorem obviously provides an easy way to generate
the semi-inverse for matrices belonging to the

We pause to observe the following:

**COROLLARY 1: **For any real-componented matrix A in the class U, the semi-inverse of A
is real-componented.

**PROOF: **A^{i} is the
image, under a polynomial with real coefficients, of A. █

Consider the well-known Pauli “particle-spin” matrices from quantum mechanics:

_{}_{} _{} _{}

We immediately have:

**COROLLARY 2: **Each of the Pauli matrices A = σ_{j}
(j = 1, 2, 3) has a semi-inverse that can be generated by the polynomial g.

**PROOF:** Each Pauli matrix has eigenvalues
λ_{1} = 1 and λ_{2} = -1, satisfying the conditions
of the theorem. █

Also, since the semi-inverse of any matrix in U can be polynomially generated, and since it is well known that when λ is an eigenvalue of a matrix A and p(x) is any polynomial it follows that p(λ) is an eigenvalue of p(A), we have:

**COROLLARY 3: ** For any matrix A in the class U (i.e. any 2X2
matrix with eigenvalues 1 and -1), the semi-inverse
has eigenvalues 1 and e^{-π}.

**PROOF:** g(1) = 1 and g(-1) =
e^{-π}. █

We may readily compute the semi-inverse of each Pauli matrix as follows:

_{}

_{}

_{}

for each of which the trace is 1 + e^{-π} in
keeping with Corollary 3.

These Pauli-matrix semi-inverses have interesting
transformational properties. For
example, while σ_{1}, viewed simply as a geometric transformation
of the plane (rather than as transition from one quantum state to another),
reflects points across the line y = x, the semi-inverse g(σ_{1})
“wraps” points close to the line. E.g.,
the topologically closed square region with vertices (0,0), (1,0), (1,1), and
(0,1) (which σ_{1} simply maps to itself since the square is
symmetric about the line y = x), when subjected to the semi-inverse
transformation g(σ_{1}), gets “folded” into a very narrow
diamond-shaped region about the line, rather like folding up an umbrella. The transformed vertices are respectively
(0,0), (½(1+e^{-π}), ½(1-e^{-π})), (1,1), and (½(1-e^{-π}),
½(1+e^{-π})).

Considering
that A^{i} for A in the matrix-class U can always be generated by the
polynomial g(x) = ½(1 – e^{-π})x + ½(1 + e^{-π}),
whose only fixed point is

_{}

the corresponding observation about the same polynomial with
a matrix-valued argument is that the only 2X2 matrix that maps (by
semi-inversion) to itself by g in the process A → A^{i} is I, the identity matrix. Thus we have:

**COROLLARY 4: **No matrix in the class U is its own
semi-inverse.

**PROOF:
**Among 2X2 g-semi-invertible matrices, only the identity matrix I is self-semi-inverse;
but I does not belong to the class U as its trace is not 0. █

Just as
matrices A^{i} (and A^{-i}) function
as semi-inverses of A and are based on exponentiation by i
and - i , the square roots of -1, one may similarly
specify such matrix powers defined by (say) the fourth roots of -1:

_{}

A matrix like A raised to the power w_{1} (in
principal value as before) might be viewed as a “demi-semi-inverse”
of A, since the operator that maps A to this power takes, when repeatedly
applied, A to its inverse A^{-1} in four stages, i.e. with three extra
“stops” along the way:

_{}

Clearly, other “fractional inverses” may be defined in terms of exponentiation by variously indexed roots of -1: the three cube roots, the eight eighth roots (producing “hemi-demi-semi-inverses”?), and so on. In the case of the Pauli matrices, it is an interesting question what exactly the implications might be, with reference to all these semi-inverses, for quantum mechanics.