Dyadic Generalized Semi-inverses of a Nonsingular Matrix
Donald R. Burleson, Ph.D.
Copyright © 2017 Donald R. Burleson. All rights reserved.
In my previous articles "The Semi-inversion Operator and the Pauli Particle-spin Matrices" (at www.blackmesapress.com/Semi-inverses.htm), "Semi-inversion of a Diagonalizable Nonsingular Matrix" (at www.blackmesapress.com/Diagonalizable.htm), and "Semi-inverting a Non-diagonalizable Matrix" (at www.blackmesapress.com/Nondiagonalizable.htm) I showed that for a nonsingular matrix A, a semi-inversion operator (here denoted ) could be defined such that and . This has a companion semi-inversion operator again having the property , the semi-inverses of A being Ai and A-i,
which in principle are easy to compute in the case that A is diagonalizable since then the necessary computations may be done by exponentiations on the eigenvalues, either by way of the canonical diagonalization
or the principal idempotent decomposition (spectral decomposition) . Even if A is not diagonalizable there are other computational methods that may avail.
Similarly, instead of employing the two square roots of -1, namely i and -i, one may define more general classes of matrix-valued operators based upon the four fourth roots of -1, the eight eighth roots of -1, etc., i.e. the dyadic roots, n-th roots of the form n = 2K for K any positive integer. For a nonsingular matrix A (whatever the computational challenges when the matrix is not diagonalizable) it is clear at least that such matrix powers can always be meaningfully defined and rather easily computed if the matrix is diagonalizable. Thus:
THEOREM: All the dyadic generalized semi-inverses of all orders exist for any diagonalizable nonsingular matrix.
PROOF: Either in canonical diagonalization form or by way of the principal idempotent decomposition (which always exists for a diagonalizable matrix), each such matrix power Aw can be computed (with details as suggested by the previous articles cited) by exponentiations on the eigenvalues, which are always nonzero when A is nonsingular.
In particular for the four fourth roots of -1, which are
and the four operators
produce matrix results which we may call "demi-semi-inverses," with patterns as outlined in this diagram, which shows the progression (through repeated applications of each operator) from A to A-1 through the intermediate demi-semi-inverse matrix forms:
Here is a similar diagram showing not only the progressions from A to A-1 but also the "return" progressions from A-1 back to A as well:
The individual "out and back" for particular operators are shown here:
Finally, this more general mapping diagram shows the effects of further applications of all the demi-semi-inverse operators to the various matrix forms, which in effect are the states in a finite-state machine:
Further groups of such operators are of course possible, producing e.g. "hemi-demi-semi-inverses" by employing the eight eighth roots of -1:
and likewise for dyadic configurations of higher order, as well as other, non-dyadic systems using e.g. the three cube roots of -1, since proof of the above theorem would equally well apply to exponentiation of the eigenvalues by any of these numbers. The dyadic cases of course have the property (as do some other similar scenarios) that the mapping configurations at each level (e.g. semi-demi-inverses for fourth roots of -1) are isomorphically embedded in the mapping configurations at all higher levels (e.g. hemi-demi-semi-inverses).