The Rotation Spectrum of a Semi-invertible Matrix

Donald R. Burleson, Ph.D.

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

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

, and have proven a theorem showing that given any

eigenvalue of a semi-invertible matrix A the corresponding eigenvalue of the (principal) semi-inverse matrix generates a rotation matrix

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

which in turn generates the rotation matrix indicated, one may regard 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 of a semi-invertible matrix A, the “stretching or contraction” rotation matrix generated by will be called the natural rotation matrix corresponding to .

Since in effect, then, each eigenvalue in the spectrum 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 produces a natural rotation matrix having the effect of rotating vectors through an angle , let the set of angles of rotation , as 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 , is precisely the set .

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

which is a rotation matrix having the angle of rotation 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 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 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 has the value while in the other case the coefficient has the value .