Multivalued Semi-inverses for Diagonalizable Nonsingular Matrices

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.

Much the same is true even for real-number eigenvalues having, as complex numbers, argument = 0.

For a real eigenvalue λ, Euler's formula again implies that one may take λⁱ as cos ln(λ) + i sin ln(λ). However,

since Euler's formula also implies that e^-2nπ = 1ⁱ for n = 0,1,2,3,..., the expression [(1)(λ)]ⁱ = (1ⁱ)(λⁱ)

is a multivalued coefficient in each term
λ_jⁱE_j = e^-2nπ[cos ln(λ_j) + i sin ln(λ_j)]E_j
of the principle idempotent expression of the semi-inverse.

The exponential factors, while remaining positive, clearly approach 0. Thus we have proven the following:

THEOREM: Every nonsingular diagonalizable matrix over the field of complex numbers
is semi-invertible and in fact has a countably infinite number of semi-inverses that form
a sequence of matrices converging to the null matrix as a limit.

COROLLARY:Every nonsingular Hermitian matrix has a countably infinite number of
semi-inverses forming a matrix sequence approaching the zero matrix as a limit.

PROOF:Every Hermitian matrix is diagonalizable.

EXAMPLE:

Let

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 principal 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 positive integer values of n.
The two results shown above are the first two elements (for n=0 and n=1) of a matrix sequence
which clearly approaches the zero matrix in the limit.