Multivalued Semi-inverses for Diagonalizable Nonsingular
Matrices having Complex Eigenvalues
Donald R. Burleson, Ph.D.
In some of my earlier research articles I have outlined procedures by which a suitable matrix A may be semi-inverted, i.e. may be subjected to a matrix-valued transform such that
In particular, I have proven that every nonsingular diagonalizable matrix is semi-invertible, since a diagonalizable matrix A always has a principal idempotent decomposition (spectral decomposition) in terms of its spectrum from which, since under the hypothesis of nonsingularity all the eigenvalues are nonzero and the function is holomorphic in any open neighborhood not containing zero, one may semi-invert the matrix by exponentiating the eigenvalues and using the resulting forms as the coefficients in the spectral decomposition of the resulting matrix:
By Euler's Formula we may then compute the exponentiated eigenvalue coefficients as
In the case of any terms in the spectral decomposition where the eigenvalue is a non-real complex number z, we may compute the logarithms in the above expression as
in principal complex value, and may call the resulting matrix the principal semi-inverse of A.
However, if we do not restrict the value of ln(z) to its principal value, admitting instead the multivalued logarithm
(where the case n = 0 gives the principal complex value result), then we have shown that a semi-inverse of A exists for each value of n, as the evaluation of the necessary coefficients in the spectral decomposition is readily managed. Thus we have shown the following:
THEOREM: Every nonsingular diagonalizable matrix, some of whose eigenvalues are nonreal complex numbers, has a countably infinite number of semi-inverses.
It is readily shown that for diagonalization a modal matrix, using respective eigenvectors as columns, is and that the resulting diagonalization is
so that we have
which in principal complex value (the case n = 0 mentioned above) gives, by Euler's Formula,
which multiplies out to give the prinicipal semi-inverse
But if, instead of the branch n = 0, we follow the branch n = 1 and write
the resulting semi-inverse, by the same procedure as described above (but this time with ), works out to be
with, similarly, other semi-inverses existing for all other integer values of n.