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algorithm for obtaining the minimal polynomial and a Jordan canonical form for an n x n matrix
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# algorithm for obtaining the minimal polynomial and a Jordan canonical form for an n x n matrix

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Written in English

### Subjects:

• Jordan matrix.,
• Polynomials.

## Book details:

Edition Notes

Includes bibliographical references.

The Physical Object ID Numbers Statement G. L. Spencer. Pagination 40 leaves : Number of Pages 40 Open Library OL22031147M

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hence distinct eigenvalues ⇒ ni = 1 ⇒ A diagonalizable dimN(λI −A) is the number of Jordan blocks with eigenvalue λ more generally, dimN(λI −A)k = X. λi=λ. min{k,ni} so from dimN(λI −A)k for k = 1,2, we can determine the sizes of the Jordan blocks associated with λ. Jordan canonical form 12–4. 6. APPLICATION OF JORDAN CANONICAL FORM A. But the lesser the degree of f(x), the better the above algorithm works. If the minimal polynomial of A is know, we take f(x) to be the minimal poly-nomial. Otherwise, we usually take f(x) to be the characteristic polynomial of A by Cayley-Hamilton. To determine r(x), we write () as () xn f(x) = q(x)+ r(x). Minimal Polynomials andJordanNormal Forms 1. Minimal Polynomials Let A be an n×n real matrix. M1. There is a polynomial p such that p(A) = 0. Proof. The space M n×n(R) of n × n real matrices is an n2-dimensional vector space over R. Take I,A,A2,,An2. There are n2 + 1 elements here, so they are linearly dependent: µ0I +µ1A+ +µ n2An 2 = 0 for some µ. Minimal Polynomial and Jordan Form Tom Leinster The idea of these notes is to provide a summary of some of the results you need for this course, as well as a di erent perspective from the lectures. Minimal Polynomial Let V be a vector space over some eld k, and let: V -V be a linear map (an ‘endomorphism of V’).File Size: KB.

A Simple Jordan Canonical Form Algorithm This document is intended for anyone who has been exposed to a second course in linear algebra and who has been mystified by the usual lengthy treatments of the Jordan canonical form and who simply wants an algorithm which can be implemented by an exact arithmetic matrix calculator such as my CMAT. minimum polynomials of A, and we will rely on the deﬁnitions of eigenvalue and eigenvector We will consider now the fundamental elements that make up the Jordan canonical form of a matrix. JORDAN BLOCKS Rewriting this we obtain (A−aI)X1 = 0 (A−aI)X2 = X1 (A−aI)X3 = X2 (A−aI)Xm = Xm−1. Notice that X1 is an eigenvector. On the computation of the Jordan canonical form of regular matrix polynomials G. Kalogeropoulos1, P. Psarrakos2 and N. Karcanias3 Dedicated to Professor Peter Lancaster on the occasion of his 75th birthday Abstract In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed. The new. How to Find Jordan Canonical ormsF Here is a method to nd a Jordan canonical form of matrices and some examples showing the method at work: Method Outline (i) orF a transformation T: V!V, use a basis to get a matrix A= [T] for the transformation. If you are just given a matrix, use that matrix. Q (ii) Compute pFile Size: KB.

characteristic polynomial Q i f i(X) and minimal polynomial f m(X). 3. Jordan Canonical Form To obtain the Jordan canonical form of T, now take kto be algebraically closed. The factorization of each ith elementary divisor polynomial, f i(X) = Q ‘ i j=1 (X ij) e ij; gives a decomposition of the ith polynomial quotient ring, k[X] hf i(X)i ˇ k File Size: KB. matrix is equivalent to a (essentially) unique Jordan matrix and we give a method to derive the latter. We also introduce the notion of minimal polynomial and we point out how to obtain it from the Jordan canonical form. Finally, we make an encounter with companion matrices. 1 Jordan form and an application Deﬁnition 1. Even if a matrix is real its Jordan normal form might be complex and we shall therefore allow all matrices to be complex. orF real matrices there is, however, a arianvt of the Jordan normal form which is real see the remarks in escThl, p. The result we want to prove is the following. Theorem 1. Let A eb an n Size: KB. 5into Jordan canonical form. 1) There is only one eigenvalue = 1 2) Nul(A (I)) = Nul(A+ I) = Span 8 Jordan canonical form of A, namely: 2 4 1 1 0 0 1 1 0 0 1 3 5 And looking at this matrix, it follows that v 1 must be an eigenvector of A, and File Size: KB.