Donald R. Burleson, Ph.D.

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


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            For a square matrix A with distinct nonzero eigenvalues λj and associated eigenvectors Vj, one may extend the relation


AxVj = λjxVj


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 λji one may make use of Euler’s identity e = cos θ + i sin θ and its implied relation ei(π + 2kπ) = -1 to determine that in principal value (i.e. with k = 0) 1i = 1 and (-1)i = e, so that more generally we have not only ai = (eln(a))i = ei 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[cos ln(b) + i sin ln (b)].  In any case we can express each λji (even when λj 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 λji; 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 AiVj = λjiVj we have (for V1)

Likewise (for V2) we have

These matrix products imply

bP + (λ1-a)Q = λ1ib

-(λ1-a)P + cQ = -λ2i1-a)


bR + (λ1-a)S = λ1i1-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(I-M)],

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

g(x) = ½(1-e)x + ½(1+e),

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(I-M)] for comparison:

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

g(x) = ½(1 – e)x + ½(1 + e).      



            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 = σj (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.

PROOF:   g(1) = 1 and g(-1) = e.  


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

for each of which the trace is 1 + e 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), ½(1-e)), (1,1), and (½(1-e), ½(1+e)).

            Considering that Ai for A in the matrix-class U can always be generated by the polynomial g(x) = ½(1 – e)x + ½(1 + e), 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.