The Rotation Spectrum of a Semi-invertible Matrix

Donald R. Burleson, Ph.D.

Copyright © 2019 by Donald R. Burleson. All rights reserved.

In earlier articles I have defined, for nonsingular matrices A, the semi-inverse operator ole.gif

such that ole1.gif (where we will denote ole2.gif in principal value as before), and the operator ole3.gif that maps a complex number z to its associated multiplication matrix

ole4.gif , and have proven a theorem showing that given any

eigenvalue ole5.gif of a semi-invertible matrix A the corresponding eigenvalue ole6.gif of the (principal) semi-inverse matrix ole7.gif generates a rotation matrix


Since it is the original A-eigenvalue ole9.gif that gives rise to the semi-inverse matrix eigenvalue

ole10.gif  which in turn generates the rotation matrix indicated, one may regard ole11.gif as the source of the rotation matrix, with the understanding that the concept we are about to define is motivated by the process of semi-inversion of the original matrix. Hence:

DEFINITION: Given an eigenvalue ole12.gif of a semi-invertible matrix A, the “stretching or contraction” rotation matrix generated by ole13.gif will be called the natural rotation matrix corresponding to ole14.gif .

Since in effect, then, each eigenvalue in the spectrum ole15.gif of a semi-invertible matrix A leads ultimately to its natural rotation matrix, based on a corresponding angle, so that in that sense a set of eigenvalues corresponds to a set of angular measures, it seems reasonable to formulate the following:

DEFINITION: For a semi-invertible matrix A, where each eigenvalue ole16.gif      produces a natural rotation matrix having the effect of rotating vectors through an angle ole17.gif , let the set of angles of rotation ole18.gif , as ole19.gif ranges over the spectrum of A, be called the rotation spectrum of A.

We immediately have, then:

LEMMA: The rotation spectrum of a semi-invertible matrix A, whose ordinary spectrum is the set of eigenvalues ole20.gif , is precisely the set ole21.gif .

PROOF: As has been shown, each ole22.gif has a natural rotation matrix

ole23.gif  which is a rotation matrix having the angle of rotation ole24.gif radians.

It is quite easy then to prove:

THEOREM: Given a semi-invertible matrix A, the sum of the angles of rotation making up the rotation spectrum of A equals the natural logarithm of the absolute value of the determinant of A.

PROOF: By the above lemma, the elements of the rotation spectrum of A are the angles of measure ole25.gif radians, and


since for any matrix the determinant is the product of the eigenvalues.

REMARKS: One should note that the relation between the ordinary spectrum of a matrix and the rotation spectrum is not one-to-one, since different eigenvalues may have the same absolute value or modulus, hence the same angle of rotation, as for example would be the case with such eigenvalues as 3 and -3, or such eigenvalues as 2 + 5i and its complex conjugate 2 - 5i, or for that matter such eigenvalues as 3 + 4i and 4 + 3i. One should also observe that even when two eigenvalues give the same angle of rotation because their absolute values or moduli are the same, the natural rotation matrices may still be different because the values of ole27.gif may be different. The complex numbers 4+ 3i and 3 + 4i, for example, have the same modulus and thus the same angle of rotation, namely ln(5) radians, but in one case the “stretching or contraction” coefficient ole28.gif has the value ole29.gif while in the other case the coefficient has the value ole30.gif .