Numerical computation of a coprime factorization of a transfer function matrix
Systems & Control Letters
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Matrix computations (3rd ed.)
SIAM Journal on Matrix Analysis and Applications
Block LU factors of generalized companion matrix pencils
Theoretical Computer Science
The Conditioning of Linearizations of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
SIAM Journal on Matrix Analysis and Applications
Vector Spaces of Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Symmetric Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Backward Error of Polynomial Eigenproblems Solved by Linearization
SIAM Journal on Matrix Analysis and Applications
Definite Matrix Polynomials and their Linearization by Definite Pencils
SIAM Journal on Matrix Analysis and Applications
A Toeplitz algorithm for polynomial J-spectral factorization
Automatica (Journal of IFAC)
Palindromic companion forms for matrix polynomials of odd degree
Journal of Computational and Applied Mathematics
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A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil—a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.