Let A be a matrix and λ0 be one of its eigenvalues having g elementary Jordan blocks in the Jordan canonical form of A. We show that for most matrices B satisfying rank (B) ≤ g, the Jordan blocks of A+B with eigenvalue λ0 are just the g− rank (B) smallest Jordan blocks of A with eigenvalue λ0. The set of matrices for which this behavior does not happen is explicitly characterized through a scal...

We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskĭı’s algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set Cmn of indecomposable canonical m× n matrices. Considering Cmn as a subset in ...

We apply the Lefschetz Fixed Point Theorem to show that every square matrix over the quaternions has right eigenvalues. We classify them and discuss some of their properties such as an analogue of Jordan canonical form and diagonalization of elements of the compact symplectic group Sp(n).

In a paper by Burke, Lewis and Overton, a first order expansion has been given for the minimum singular value of A−zI, z ∈ C, about a nonderogatory eigenvalue λ of A ∈ Cn×n. This note investigates the relationship of the expansion with the Jordan canonical form of A. Furthermore, formulas for the condition number of eigenvalues are derived from the expansion.

We consider a key case in the fundamental and substantial problem of the possible Jordan canonical forms of A, B,C ∈Mn(F ) when C = AB. If A ∈ M2k(F ) (respectively B, C ∈ M2k(F ) ) is diagonalizable with two distinct eigenvalues a1, a2 (respectively b1, b2, and c1, c2), each with multiplicity k, and when C = AB, all possibilities for a1, a2, b1, b2, c1, c2 are characterized. The possibilities ...

In this paper, we introduce the notions of weakly normal and normal matrix polynomials, with nonsingular leading coefficients. We characterize these matrix polynomials, using orthonormal systems of eigenvectors and normal eigenvalues. We also study the conditioning of the eigenvalue problem of a normal matrix polynomial, constructing an appropriate Jordan canonical form.

In a paper by Burke, Lewis and Overton, a first order expansion has been given for the minimum singular value of A−zI, z ∈ C, about a nonderogatory eigenvalue λ of A ∈ Cn×n. This note investigates the relationship of the expansion with the Jordan canonical form of A. Furthermore, formulas for the condition number of eigenvalues are derived from the expansion.

Definition 2.1. Suppose p1, . . . , pk are polynomials in F[x] which are not all 0. Set I = 〈p1, . . . , pk〉. Let d denote the monic generator of I. We call d the greatest common divisor of the pi and write d = gcd(p1, . . . , pk). If d = 1, we say that the polynomials p1, . . . , pk is relatively prime. Note that, by definition, there exists polynomials q1, . . . , qk ∈ F[x] such that d = p1q1...

This paper outlines a proof of the Jordan Normal Form Theorem. First we show that a complex, finite dimensional vector space can be decomposed into a direct sum of invariant subspaces. Then, using induction, we show the Jordan Normal Form is represented by several cyclic, nilpotent matrices each plus an eigenvalue times the identity matrix – these are the Jordan