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 ole.gif belonging to the spectrum of the matrix A, for reasons having to do with what I have defined as the principal semi-inverse ole1.gif and the corresponding eigenvalue

ole2.gif  I have previously defined the natural rotation matrix generated by ole3.gif to be the matrix


ole4.gif


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: ole5.gif

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:


ole6.gif


ole7.gif ole8.gif where the elements of this product matrix are


ole9.gif


ole10.gif


ole11.gif


ole12.gif


Thus the desired matrix comes out as


ole13.gif


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.