A Jordan canonical form for reachable linear systems

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, . . ....

Jordan Canonical Form

is the geometric multiplicity of λk which is also the number of Jordan blocks corresponding to λk . • The orders of the Jordan Blocks of λk must sum to the algebraic multiplicity of λk . • The number of Jordan blocks corresponding to an eigenvalue λk is its geometric multiplicity. • The matrix A is diagonalizable if and only if, for any eigenvalue λ of A , its geometric and algebraic multiplici...

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 . . . . . . . . . . . . . ....

Enhancing the Jordan canonical form

The Jordan canonical form parametrises similarity classes in the nilpotent cone Nn, consisting of n× n nilpotent complex matrices, by partitions of n. Achar and Henderson (2008) extended this and other well-known results about Nn to the case of the enhanced nilpotent cone C ×Nn. 1. Jordan canonical form The Jordan canonical form (JCF), introduced in 1870 [10], is one of the most useful tools in...

Lecture Notes for 416: Jordan Canonical Form