Irrotational Eigenvalues of Natural Rotation Matrices

Donald R. Burleson, Ph.D.

Copyright © 2020 by Donald R. Burleson. All rights reserved.

As described in my previous research articles, if A is a diagonalizable nonsingular (and therefore semi-invertible) matrix, and if ole.gif is any eigenvalue of A, the natural rotation matrix associated with this eigenvalue is


motivated by the fact that if we designate the matrix ole2.gif to be the “associated multiplication matrix” for a complex number x + yi, then given ole3.gif and the associated eigenvalue ole4.gif of the principal semi-inverse of A, the associated multiplication matrix for this latter eigenvalue is the natural rotation matrix indicated, the exponential coefficient and the absolute value (or modulus) bars arising from the general case where the eigenvalue ole5.gif is a not-necessarily-real complex number.

We may routinely determine the eigenvalues of the natural rotation matrix ole6.gif as follows:


the zeros of which are


so that by Euler’s Formula we have shown that:

LEMMA: For a diagonalizable nonsingular (and thus semi-invertible) matrix A, one of whose eigenvalues is ole9.gif , the eigenvalues of the natural rotation matrix ole10.gif generated by this eigenvalue are ole11.gif

It follows that, to carry the process one step further, the eigenvalues of the natural rotation matrix generated by

ole12.gif  and its reciprocal 1

(in principal complex value), and likewise for the other eigenvalue ole13.gif

We may ask: what is the natural rotation matrix generated by ole14.gif ? By definition,



A similar computation shows that for the other eigenvalue the natural rotation matrix is


Thus each of these natural rotation matrices reduces to a nonzero constant multiple of the identity matrix, which of course only “rotates” a vector through an angle of 0 radians, as well as possibly altering its length. This motivates the

DEFINITION: For a nonzero eigenvalue of a matrix, if the associated natural rotation matrix reduces to a nonzero constant multiple of the 2X2 identity matrix, the eigenvalue will be said to be an irrotational eigenvalue.

In particular, the two eigenvalues just examined, having modulus 1, work out to be irrotational. Gathering all the observations made above, we have established the

THEOREM: For any nonzero eigenvalue ole18.gif of a matrix, and in particular for any eigenvalue of a diagonalizable and nonsingular (hence semi-invertible) matrix, ole19.gif generates a natural rotation matrix whose own eigenvalues ole20.gif and ole21.gif are irrotational.

With this result we may also readily prove the following:

THEOREM: Let A be a diagonalizable nonsingular (hence semi-invertible) matrix. Let ole22.gif be any eigenvalue of A, and let ole23.gif the natural rotation matrix generated by ole24.gif . The inverse matrix ole25.gif has an eigenvalue whose natural rotation matrix has the same (irrotational) eigenvalues as ole26.gif , but the two natural rotation matrices give opposite angles of rotation.

PROOF: It is well known that for any eigenvalue ole27.gif of a nonsingular matrix A, the number ole28.gif is an eigenvalue of ole29.gif .                          Indeed if spectrum(A) = ole30.gif then

ole31.gif  Since the eigenvalues of ole32.gif have been shown to be ole33.gif it follows that the eigenvalues of ole34.gif are


the same as for ole36.gif

And since the angle of rotation for ole37.gif the angle of rotation for


REMARK: Since the eigenvalues of A and its inverse occur in mutually reciprocal pairs, in which each pair leads to two natural rotation matrices with opposite angles of rotation, the sum of the angles of rotation of all the natural rotation matrices generated by all the eigenvalues of A and its inverse taken together is 0.