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STRONGLY MINIMAL SELF-CONJUGATE LINEARIZATIONS
FOR POLYNOMIAL AND RATIONAL MATRICES

PAUL VAN DOOREN

We prove that we can always construct strongly minimal linearizations of an arbitrary
rational matrix from its Laurent expansion around the point at inﬁnity, which happens
to be the case for polynomial matrices expressed in the monomial basis. If the rational
matrix has a particular self-conjugate structure we show how to construct strongly
minimal linearizations that preserve it. The structures that are considered are the
Hermitian and skew-Hermitian rational matrices with respect to the real line, and the
para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis.
We pay special attention to the construction of strongly minimal linearizations for the
particular case of structured polynomial matrices. The proposed constructions lead to
eﬃcient numerical algorithms for constructing strongly minimal linearizations. The fact
that they are valid for any rational matrix is an improvement on any other previous
approach for constructing other classes of structure preserving linearizations, which
are not valid for any structured rational or polynomial matrix. The use of the
recent concept of strongly minimal linearization is the key for getting such
generality.

Strongly minimal linearizations are Rosenbrock’s polynomial system matrices of the
given rational matrix, but with a quadruple of linear polynomial matrices (i.e. pencils) : \[ L(\lambda ):=\left [\begin{array}{ccc} A(\lambda ) & -B(\lambda ) \\ C(\lambda ) & D(\lambda ) \end{array}\right ], \]
where \(A(\lambda )\) is regular, and the pencils \( \left [\begin{array}{ccc} A(\lambda ) & -B(\lambda ) \end{array}\right ]\) and \( \left [\begin{array}{ccc} A(\lambda ) \\ C(\lambda ) \end{array}\right ]\) have no ﬁnite or inﬁnite eigenvalues. Strongly
minimal linearizations contain the complete information about the zeros, poles and
minimal indices of the rational matrix and allow to recover very easily its eigenvectors
and minimal bases. Thus, they can be combined with algorithms for the generalized
eigenvalue problem for computing the complete spectral information of the rational
matrix.

Our results are inspired by the work of Israel Gohberg and his coauthors.

This is joint work with Froilán M. Dopico and María C. Quintana

Université catholique de Louvain

Email address: vandooren.p@gmail.com