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 is any eigenvalue of A, the natural rotation matrix associated with this eigenvalue is

motivated by the fact that if we designate the matrix to be the “associated multiplication matrix” for a complex number x + yi, then given and the associated eigenvalue 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 is a not-necessarily-real complex number.

We may routinely determine the eigenvalues of the natural rotation matrix 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 , the eigenvalues of the natural rotation matrix generated by this eigenvalue are

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

and its reciprocal 1

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

We may ask: what is the natural rotation matrix generated by ? 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 of a matrix, and in particular for any eigenvalue of a diagonalizable and nonsingular (hence semi-invertible) matrix, generates a natural rotation matrix whose own eigenvalues and are irrotational.

With this result we may also readily prove the following:

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

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

Since the eigenvalues of have been shown to be it follows that the eigenvalues of are

the same as for

And since the angle of rotation for 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.