THE SEMI-SQUARE OPERATOR AND SIMILAR MATRIX-VALUED

OPERATORS ON THE SPECTRUM-EQUIVALENCE CLASS

CONTAINING THE PAULI MATRICES, AS SPECIAL CASES OF

A MORE GENERAL THEOREM ABOUT MATRIX-POWER TRANSFORMS

Donald R. Burleson, Ph.D.

December 2006

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

(To go to the "About the Author" page, click here.)

The purpose of this paper is to show that for the class of matrices including the Pauli particle-spin matrices in quantum theory a number of interesting transforms (two of which I call the semi-inversion operator and the semi-square operator) are possible, and that the computations involved may be greatly simplified in view of a more general theorem, special cases of which lead to the matrix-valued operators mentioned. I will present the more general theorem first.

THEOREM: For any 2-by-2 matrix A with eigenvalues λ₁ ≠ λ₂ and for any complex number z,

A^z = [ (λ₁^z – λ₂^z)A + (λ₁λ₂^z – λ₂λ₁^z)I ]

where A^z is the matrix defined by principal complex-valued exponentiations λ_j^z on the eigenvalues in the relation

A^zv_j = λ_j^zv_j (j = 1, 2)

for representative eigenvectors v_j corresponding respectively to the eigenvalues λ_j, provided the eigenvalue exponentiations do not lead to division by zero.

PROOF: Let A = with distinct eigenvalues λ₁ and λ₂. We first consider the case b ≠ 0. To determine eigenvectors (x y)^T we let

= and = .

These equations imply representative eigenvectors

and

so that for the diagonalizing similarity relation A = MM^-1 the modal matrix and its inverse are

and .

Thus we have

A^z =

Since (l ₁-a)(l ₂-a) = l _{1l 2} – (l ₁+l ₂)a + a² as the characteristic polynomial of A is equal (with det A = ab – bd = l _{1l 2} and trace A = a + d = l ₁ + l ₂) to ad – bc – a² – ad + a² = -bc, the matrix in question equals

In this matrix, the element in row 2, column 2 is equal (since, again, a + d = l ₁ + l ₂, the trace of A) to l ₁^z(l ₂ – d) + l ₂^z(d - l ₁) = d(l ₂^z - l ₁^z) + l ₁^zl ₂ - l ₂^zl ₁ so that the matrix may be written as

as needed, for the case b ¹ 0.

As for the remaining case, if b = 0 then the matrix A is triangular, with its eigenvalues on the diagonal, so in this case it can be written as

.

To determine eigenvectors (x y)^T we again examine the relations A(x y)^T = λ_j(x y)^T to find that this time the representative eigenvectors may be taken to have the form

and

so that the modal matrix M in the similarity relation

is

with inverse

Then

A^z =

as required to establish the case b = 0. This completes the proof. █

Among many other things, this theorem provides an easy way to compute matrix powers A^z when z is a real integer. For example, for

with eigenvalues λ₁ = 5 and λ₂ = 2,

we may readily compute

However, the most interesting special cases are those involving other kinds of exponents z.

THE SEMI-INVERSION OPERATOR

When z = i, the theorem implies what I call the semi-inversion operator, so called because (Aⁱ)ⁱ = A^-1, the regular multiplicative inverse of A:

(It is assumed throughout that for complex exponentiations on the eigenvalues, where multivalued functions are concerned, we choose principal complex value.) In particular, for the class U of 2-by-2 matrices having eigenvalues 1 and –1, i.e. the spectrum-equivalence class containing the Pauli matrices

from quantum theory (the matrices giving vector-component projections, along various axes, of spin angular momentum), we have:

COROLLARY 1: The (principal) semi-inverse of any matrix A belonging to the class U (of two-by-two matrices having trace 0 and determinant –1) is given by evaluating, at A, the first-degree polynomial function

PROOF: This follows immediately from the main theorem, as 1ⁱ = 1 and (-1)ⁱ = e^-π in principal complex value. █

(For a more detailed discussion of the semi-inversion operator on the spectrum-equivalence class containing the Pauli matrices, click here.)

THE SEMI-SQUARE OPERATOR

With z = , the main theorem implies an operator that I call the semi-square operator, since

The (principal) semi-square of A, then, is computed by

By this method one may compute, for example (using eigenvalues λ₁ = 4 and λ₂ = 1):

,

a result that is component-wise between

and .

In particular, as before, if A is a matrix spectrum-equivalent to the Pauli matrices, with eigenvalues 1 and –1, then we have:

COROLLARY 2: For A any matrix in the class U of 2-by-2 matrices with trace 0 and determinant –1, the (principal) semi-square of A may be computed by evaluating, at A, the first-degree polynomial function

PROOF: The eigenvalues of A are 1 and –1, and in principal complex value we have

so that the corollary follows immediately from the main theorem. █

Clearly one may also use the above techniques to compute such results as

(the principal "semi-cube" of A), as well as

(the "demi-semi cubes" of A), etc.

as well as an infinitude of other matrix-valued operators based upon complex exponentiations, e.g. exponentiations by the nth roots of –1 to produce arbitrarily protracted chains of matrices "between" A and A^-1 to generalize the notion of the semi-inverse into "hemi-demi-semi-inverses" and so on.

The remarkable fact is that all these matrix forms are generated by first-degree matrix-argumented polynomial functions in which the coefficients are simple functions of the eigenvalues. It is precisely this sort of thing that makes eigenvalues central to matrix theory. I conclude with one other quick result:

COROLLARY 3: For all the above matrix transforms A^z the eigenvalues are λ₁^z and λ₂^z.

PROOF: Since λ being an eigenvalue of a matrix A implies, for any polynomial f(x), that f(λ) is an eigenvalue of f(A), one merely applies this idea to the generating function

for which, as we may verify by direct computation, h(λ_j) = λ_j^z, j = 1, 2. █