## Jordan Canonical Form: Theory and PracticeJordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T: V → V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1. We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J, and a refinement, the labeled eigenstructure picture (ℓESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader. Table of Contents: Fundamentals on Vector Spaces and Linear Transformations / The Structure of a Linear Transformation / An Algorithm for Jordan Canonical Form and Jordan Basis |

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alg-mult(A algorithm basis of Cn basis ofV BDBUTCD form block diagonal matrix block upper triangular block with diagonal cA{x Cayley-Hamilton Theorem chain of length change-of-basis matrix choose i>2 choose U3 cnvn columns complex numbers constant diagonal Corollary deﬁne Deﬁnition diagonal entries equal diagonalizable dimension dimensional with basis eigenspace eigenvalue eigenvector of index Ej(k Ej(X equivalent Example ﬁnd find the labels ﬁnding ﬁrst Fj+1 geom-mult(3 geometric multiplicity Given the bases IESP invariant subspace invertible matrix irreducible factor jmax Jordan basis Jordan block Jordan Canonical Form JP~l Lemma length 1 associated lESP Let W1 linear transformation linearly independent mA(x matrix in Jordan matrix with constant matrix with diagonal minimum polynomial monic polynomial n-by-n matrix nodes at height old nodes P J P~l Pc^B polynomial of degree Proof Proposition Remark scalar matrix set of vectors space and let square matrix subspace of Cn T-invariant TA(v upper triangular matrix vector space