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.
© 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,
Az = [ (λ1z – λ2z)A + (λ1λ2z – λ2λ1z)I ]
where Az is the matrix defined by principal complex-valued exponentiations λjz on the eigenvalues in the relation
Azvj = λjzvj (j = 1, 2)
for representative eigenvectors vj corresponding respectively to the eigenvalues λj, 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
so that for the diagonalizing similarity relation A = MM-1 the modal matrix and its inverse are
Thus we have
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 1z(l 2 – d) + l 2z(d - l 1) = d(l 2z - l 1z) + l 1zl 2 - l 2zl 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 = λj(x y)T to find that this time the representative eigenvectors may be taken to have the form
so that the modal matrix M in the similarity relation
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 Az 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
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 Az the eigenvalues are λ1z and λ2z.
PROOF: Since λ being an eigenvalue of a matrix A implies, for any polynomial f(x), that f(λ) is an eigenvalue off(A), one merely applies this idea to the generating function
for which, as we may verify by direct computation, h(λj) = λjz, j = 1, 2. █