Products of Natural Rotation Matrices

Donald R. Burleson, Ph.D.

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

Given any nXn diagonalizable nonsingular (therefore semi-invertible) matrix A, and given any eigenvalue belonging to the spectrum of the matrix A, for reasons having to do with what I have defined as the principal semi-inverse and the corresponding eigenvalue

I have previously defined the *natural rotation matrix*
generated by
to be the matrix

It is now a straightforward matter to prove the following

THEOREM: The product of the natural rotation matrices for any set of eigenvalues of an nXn diagonalizable nonsingular (thus semi-invertible) matrix A is equal to the natural rotation matrix of the product of those eigenvalues:

In particular, the product of the natural rotation matrices generated by all the eigenvalues in the spectrum of the given matrix (employing each eigenvalue as many times as its algebraic multiplicity requires) is equal to the natural rotation matrix of det(A).

PROOF: We begin with the case of two eigenvalues:

where the elements of this product matrix are

Thus the desired matrix comes out as

This result readily generalizes to more than two eigenvalues by mathematical induction. It only remains to be pointed out that when one includes all the eigenvalues of the spectrum of matrix A, the product of their natural rotation matrices equals the natural rotation matrix of det(A), since the product of the eigenvalues equals the determinant of A. This completes the proof.