Kronecker's canonical form and the QZ algorithm

Kronecker's Canonical Form and the Qz Algorithm By

In the Q,Z algorithm the eigenvalues of Ax = ABx are computed via a reduction to the form & = AEx where A" and E are upper triangular. The eigenvalues are given by hi = aii/bii . It is shown that when the pencil A" hg is singular or nearly -_ N singular a value of hi may have no significance even when aii and cii are of full size.

A Rational Canonical Form Algorithm

The Jordan Canonical Form

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....

The Canonical Contact Form

The structure group and the involutive differential system that characterize the pseudo-group of contact transformations on a jet space are determined.

The Jordan canonical form

4 The minimal polynomial of a linear transformation 7 4.1 Existence of the minimal polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 The minimal polynomial for algebraically closed fields . . . . . . . . . . . . . . . . . 8 4.3 The characteristic polynomial and the Cayley–Hamilton theorem . . . . . . . . . . . 9 4.4 Finding the minimal polynomial . . . . . . . . . . . . . ....