Matrix Semi-inversion and Complex Number Semi-Reciprocation

Donald R. Burleson, Ph.D.

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)].