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^xV_j = λ_j^xV_j

to complex values of x, and in particular to x = i, to define (in terms of principal complex value) the matrix power Aⁱ. For the necessary eigenvalue powers λ_jⁱ 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) 1ⁱ = 1 and (-1)ⁱ = e^-π, so that more generally we have not only aⁱ = (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), aⁱ = (-b)ⁱ = (-1)ⁱbⁱ = e^-π[cos ln(b) + i sin ln (b)]. In any case we can express each λ_jⁱ (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 Aⁱ.

The curious thing about Aⁱ is that since (Aⁱ)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) = Aⁱ as the (principal) semi-inversion operator, and may designate the image matrix Aⁱ as the (principal) semi-inverse of A. Since applying this operator twice in a row takes A to A^-1, the semi-inverse Aⁱ 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 Aⁱ, and both Aⁱ and A^-i are multivalued due to the multivalued nature of the eigenvalue exponentiations λ_jⁱ; but the term “semi-inverse” herein assumes principal-valued entities throughout and refers to the principal-valued matrix Aⁱ 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 aⁱ = 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 λ₁ and λ₂; 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 V₁ = (x,y)^T is an eigenvector belonging to the eigenspace corresponding to λ₁, we have

whence ax + by = λ₁x

from which we may infer an eigenvector of the form

.

Likewise from

whence cx + dy = λ₂y

we may infer an eigenvector of the form

since trace(A) = a + d = λ₁ + λ₂. Now let

Then from the relation AⁱV_j = λ_jⁱV_j we have (for V₁)

Likewise (for V₂) we have

These matrix products imply

bP + (λ₁-a)Q = λ₁ⁱb

-(λ₁-a)P + cQ = -λ₂ⁱ(λ₁-a)

and

bR + (λ₁-a)S = λ₁ⁱ(λ₁-a)

-(λ₁-a)R + cS = λ₂ⁱc.

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 λ₁-a, we may always write d – λ₂, and since the eigenvalues λ₁ and λ₂ 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 = λ₁ and d = λ₂, 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) = Aⁱ 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 λ₁ = 1 and λ₂ = -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

Mⁱ = 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

Mⁱ = g(M).

PROOF: Let

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

a² + 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 class U.

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: Aⁱ 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 λ₁ = 1 and λ₂ = -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 σ₁, 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(σ₁) “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 σ₁ simply maps to itself since the square is symmetric about the line y = x), when subjected to the semi-inverse transformation g(σ₁), 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 Aⁱ 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 → Aⁱ 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 Aⁱ (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 w₁ (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.