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Jordan Canonical Form - Steven H. Weintraub - häftad - Adlibris

Oct 26, 2005 This matrix B is called the Jordan canonical form of the matrix A. If the eigenvalues of A are real, the matrix B can be chosen to be real. If some  Example of Jordan canonical form with dimension 3, in this case, the eigenspace corresponding to the only eigenvalue, has dimension 2, so we have to  You can think of Jordan cannonical form as a generalization of the the concept of a "diagonal matrix". It's easy to to multiplications and find  Jordan Block Matrices. An m×m matrix is of the Jordan block form if it has a constant on the principal diagonal and 1's for all the elements next to the principal  Oct 3, 2007 Brualdi, Richard A., The Jordan canonical form: an old proof. Amer. Math.

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We first develop JCF, including the concepts involved in it eigenvalues, eigenvectors, and chains of generalized eigenvectors. The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a nite-dimensional vector space over an al-gebraically closed eld. Here we develop it using only the most basic concepts of linear algebra, with no reference to determinants or ideals of polynomials. THEOREM 1. Let 1, :::, Computing the Jordan Canonical Form Let A be an n by n square matrix. If its characteristic equation χ A(t) = 0 has a repeated root then A may not be diagonalizable, so we need the Jordan Canonical Form. Suppose λ is an eigenvalue of A, with multiplicity r as a root of χ A(t) = 0.

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In this section will progress from the specialized case of a nilpotent matrix to the totally general case of any square matrix. It is known that the Jordan canonical form of a matrix is not a continuous function of its elements, although the eigenvaluesof a matrix are continuousfunctions[GVL96, problems7.1.5–6].

Jordan canonical form

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What is the ‘nicest’ form we can get an arbitrary matrix into. We already know that, to every eigenvalue, there is a corresponding eigenvector. If an nxn matrix has n linearly independent eigenvectors, then it is diagonalizable.

(Ayres 1962, p. 206).
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We show how to find a b We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. We demonstrate this with an example and provide several exercises.

We demonstrate this with an example and provide several exercises. JORDAN CANONICAL FORMS 3 2.
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The Jordan Canonical Form { Examples Example 1: Given A= 0 1 1 2 ; nd its JCF and P. Here: ch A(t) = (t+ 1)2 A( 1) = 1))J= 1 1 0 1 : We want to nd an invertible matrix Psuch that (1) P 1AP = Jor, equivalently, AP = PJ: Write P= (~v 1 j~v 2). Then, since AP= (A~v 1 jA~v 2) and PJ= (~v 1 j~v 1 ~v 2), we see that equation (1) is equivalent to: 1) A~v 1 = ~v 1 or (A+ I)~v 1 =~0 its blocks are Jordan blocks; in other words, that A= UBU 1, for some invertible U. We say that any such matrix Ahas been written in Jordan canonical form.

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The Jordan canonical form (Jordan normal form) results from attempts to convert a matrix to its diagonal form by a similarity transformation. For a given matrix A, find a nonsingular matrix V, so that inv(V)*A*V, or, more succinctly, J = V\A*V, is “as close to diagonal as possible.” Jordan Canonical Form. The block diagonal system having the eigenvalues along the diagonal and ones in some of the superdiagonal elements (which serve to couple repeated eigenvalues) is called Jordan canonical form. Each block size corresponds to the multiplicity of the repeated pole.

This is left as an exercise. Problems of Jordan Canonical Form. From introductory exercise problems to linear algebra exam problems from various universities. Basic to advanced level. Jordan Canonical Form Recall the following definition: Definition 1.