NONTRIVIAL INVARIANT SUBSPACES OF LINEAR OPERATOR PENCILS

Linear Algebra: Invariant Subspaces

Introduction 1 1. Invariant Subspaces 3 2. Eigenvectors, Eigenvalues and Eigenspaces 11 3. Cyclic Spaces 14 4. Prime and Primary Vectors 15 5. The Cyclic Decomposition Theorem 20 6. Rational and Jordan Canonical Forms 22 7. Similarity 23 8. The Cayley-Hamilton Polynomial (Or: Up With Determinants?) 24 9. Extending The Ground Field 25 9.1. Some Invariances Under Base Extension 25 9.2. Semisimpli...

The Stability Radius of Linear Operator Pencils

Let T and S be two bounded linear operators from Banach spaces X into Y and suppose that T is Fredholm and the stability number k(T ;S) is 0. Let d(T ;S) be the supremum of all r > 0 such that dimN(T − λS) and codim R(T − λS) are constant for all λ with |λ| < r. It was proved in 1980 by H. Bart and D.C. Lay that d(T ;S) = limn→∞ γn(T ;S) , where γn(T ;S) are some non-negative (extended) real nu...

Invariant Subspaces of Voiculescu's Circular Operator

The number of invariant subspaces under a linear operator on finite vector spaces

Let V be an n-dimensional vector space over the finite field Fq and T a linear operator on V . For each k ∈ {1, . . . , n} we determine the number of k-dimensional T -invariant subspaces of V . Finally, this method is applied for the enumeration of all monomially nonisometric linear (n, k)-codes over Fq . 0. Introduction Let q be a power of a prime p, Fq the finite field with q elements and n a...

Linear and Nonlinear Controllability through Invariant Subspaces

This paper has the following aims: (i) to provide dynamical interpretations of the rank conditions for controllability and stabilizabil-ity of linear control systems and try to understand what they mean geometrically; (ii) to re-visit the theory of canonical forms from an algebraic viewpoint; (iii) to give some reened genericity results on controllable pairs; (iv) to make precise the way in whi...