An Arithmetic for Matrix Pencils: Theory and New Algorithms

Abstract

This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case. These operations give an unusual perspective on a variety of pencil related computations. We derive generalizations of monodromy matrices and the matrix exponential. A new algorithm for computing a pencil-arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces.

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

“…Pencil arithmetic is a technique to extend several common matrix operations, such as matrix sum and matrix product to matrix pencils. One of its many uses is to perform linear algebra operations on matrices of the form , while storing and operating on the pair .…”

Permuted Graph Matrices and Their Applications

“…This modified version is called doubling algorithm; it was introduced (with a different derivation) for unstructured invariant subspace problems without the use of permuted graph bases in [4,33], and for symplectic problems with the special choice S D I (graph basis without permutation) in [2,11,12,29], and then generalized to make full use of permuted graph bases in [35]. The algorithm that we described first is known as inverse-free sign method; it appeared without the use of permuted graph bases in [6], then with S D I in [20], and with permuted graph bases in [36]. Permuted graph bases are important here because they ensure that the iterative procedure produces an exactly Hamiltonian (or symplectic) pencil at each step and steers clear of numerically singular pencils.…”

“…The work [6] contains a general theory of operations with matrix pencils that describes how to produce "matrix pencil versions" of rational functions, such as (5.8) and (5.9) for f .z/ and g.z/. This modified version is called doubling algorithm; it was introduced (with a different derivation) for unstructured invariant subspace problems without the use of permuted graph bases in [4,33], and for symplectic problems with the special choice S D I (graph basis without permutation) in [2,11,12,29], and then generalized to make full use of permuted graph bases in [35].…”

Linear relations and the Kronecker canonical form

“…The kernel representation (3.6) of a linear relation A was already considered in [4,5] using the notation…”

“…In the sense of linear relations, the inverse E −1 of a non-invertible matrix E is given as the subspace of all tuples ( Ex x ) in C n × C n . Then expression (1.2) has a natural meaning, E −1 A = { (x, y) ∈ C n × C n | Ax = Ey } , which was already studied in [4,5]. An eigenvector at λ ∈ C of E −1 A is a tuple of the form (x, λx) ∈ E −1 A, x = 0, and thus satisfies Ax = λEx.…”