Matrix Semi-inversion and Complex Number Semi-Reciprocation
Donald R. Burleson, Ph.D.
Copyright © 2019 by Donald R. Burleson. All rights reserved.
In earlier articles I have defined and described what may be called the (principal) semi-inversion operator for diagonalizable nonsingular matrices A, such that , where the principal value of is denoted . (In the special case where A = (a) is one-by-one, we may call a semi-inverse of A simply a semi-reciprocal.) Let us now look at a particular matter related to this matrix-valued transform, with respect to complex number products.
If z = x + yi is any complex number, we will define the associated multiplication matrix of z as the matrix
The motivation for this is the fact that this matrix form provides a convenient way to multiply complex numbers, in that
(x+yi)(u + vi) can be expressed as
For example, (3 + 5i)(2 - 7i) = 41 - 11i can be computed as
One may easily prove the necessary
LEMMA: For any nonzero complex number z = x + yi , the eigenvalues of the matrix
are with respective eigenvectors
PROOF: Let A = The characteristic polynomial of A is
the zeros of which are
We also observe that
showing that the eigenvectors are as stated, regardless of what complex number z is.
It follows that for purposes of diagonalization of A = the necessary modal matrix (using the eigenvectors as columns) is
giving the canonical diagonalization as
Since the function is holomorphic in any open neighborhood in the complex plane not containing the origin, we may semi-invert A by writing
and this multiplies out to give
On inspection one sees that this is the associated multiplication matrix for the complex number
which proves the
THEOREM: The (principal) semi-inverse of the associated multiplication matrix of a nonzero complex number z is the associated multiplication matrix of the (principal) semi-reciprocal of z. That is to say: φ[μ(z)] = μ[φ(z)].