Session B7 - Numerical Linear Algebra
July 14, 15:30 ~ 16:00
Vector Spaces of Linearizations for Rectangular Matrix Polynomials
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI, INDIA - firstname.lastname@example.org
The seminal work [MMMM06] introduced vector spaces of matrix pencils, with the property that almost all the pencils in the vector space are strong linearizations of a given square regular matrix polynomial. This work was subsequently extended to include the case of square singular matrix polynomials in [DTDM09]. Extending these ideas, we construct similar vector spaces of rectangular matrix pencils such that almost every matrix pencil of the space is a strong linearization of a given rectangular matrix polynomial $P$ in a generalized sense. Moreover, the minimal indices of $P$ can be recovered from those of the matrix pencil.
We further show that such pencils can be `trimmed' to form smaller pencils that are unimodular equivalent to $P.$ The resulting pencils are almost always strong linearizations of $P.$ Moreover they are easier to construct and are often smaller than the Fiedler linearizations of $P$ introduced in [DTDM12]. Further, the backward error analysis carried out in [DLPVD16] when applied to these trimmed linearizations shows that under suitable conditions, the computed eigenstructure of the linearizations obtained from some backward stable algorithm yield the exact eigenstructure of a slightly perturbed matrix polynomial.
[DTDM09] F. De Teran, F. M. Dopico, and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electron. J. Linear Algebra, 18(2009), pp. 371-402.
[DTDM12] F. De Teran, F. M. Dopico, and D. S. Mackey, Fiedler companion linearizations for rectangular matrix polynomials, Linear Algebra Appl., 437(2012), pp. 957-991.
[DLPVD16] F. M. Dopico, P. W. Lawrence, J. Perez and P. Van Dooren, Block Kronecker linearizations of matrix polynomials and their backward errors, MIMS Eprint, 2016.34, 2016.
[MMMM06] D. S. Mackey, N. Mackey, C. Mehl and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Annal. Appl., 28(4): 971-1004, 2006.
Joint work with Biswajit Das (Indian Institute of Technology Guwahati).