Jordan

# Jordan - Linear Algebra Canonical Forms re Friedberg Insel and Spence Linear Algebra 2nd ed Prentice-Hall(Chapter 7 p r Introduction The advantage

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Canonical Forms r Friedberg, Insel, and Spence, “Linear Algebra”, 2nd ed., Prentice-Hall. (Chapter 7) && &&&p ± r& & Linear Algebra

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Jordan_2 Introduction The advantage of a diagonalizable linear operator is the simplicity of its description. Not every linear operator is diagonalizable. Example: T : P 2 P 2 with T ( f ) = f’ , the derivative of f . The matrix of T with respect to the standard basis {1, x , x 2 } for P 2 is A = The characteristic polynomial of A is A has only one eigenvalue ( λ =0)with multiplicity 3. The eigenspace corresponding to λ =0 is { | r R } A is not diagonalizable 0 0 0 2 0 0 0 1 0 3 0 0 2 0 0 1 λ - = - - - 0 0 r
Jordan_3 The purpose of this chapter is to consider alternative matrix representations for nondiagonalizable operators. These representations are called canonical forms . There are different kinds of canonical forms, there advantages and disadvantages depend on how they applied. Our focus Jordan canonical form This form is always available if the underlying field is algebraically closed, that is, if every polynomial with coefficients from the field splits.

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Jordan_4 7.1 General Eigenvectors Let T : V V be a linear operator, A be a matrix representation of T . Suppose A has eigenvalues λ 1 , λ 2 , …, λ n , with has n corresponding, linearly independent eigenvectors v 1 , v 2 , …, v n . Let B ={ x 1 , x 2 , …, x n } (note that B is a basis for V ), then = n B T λ 0 0 0 0 0 0 ] [ 2 1 where [ T ] B is the diagonal matrix representation of T . Since A may be not diagonalizable, we will prove that:
Jordan_5 For any linear operator whose characteristic polynomial splits, i.e., characteristic polynomial= there exists an ordered basis B for V such that ) ( ) )( ( 2 1 n x x x λ - - - = k B J J J T 0 0 0 0 0 0 ] [ 2 1 for some eigenvalue λ j of T . Such a matrix J i is called a Jordan block corresponding to λ j , and the matrix [ T ] B is called the Jordan canonical form of T . The basis B is called a Jordan canonical basis for T. where J i is a square matrix of the form [ λ j ] or the form j j j j 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1

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Jordan_6 Example 1 The 8 × 8 matrix is a Jordan canonical form of a linear operator T : C 8 C 8 ; that is, there exists a basis B ={ x 1 , x 2 , …, x 8 } for C 8 such that [ T ] B = J . = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 1 2 J Note that the characteristic polynomial for T and J is det( J I) = ( λ- 2) 4 ( 3) 2 λ 2 , and only x 1 , x 4 , x 5 , x 7 of B ={ x 1 , x 2 , …, x 8 } are eigenvectors of T .
Jordan_7 It will be proved that every operator whose characteristic polynomial splits has a unique Jordan canonical form (up to the order of the Jordan blocks).

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## This note was uploaded on 09/12/2011 for the course E 101 taught by Professor J during the Spring '11 term at Adrian College.

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Jordan - Linear Algebra Canonical Forms re Friedberg Insel and Spence Linear Algebra 2nd ed Prentice-Hall(Chapter 7 p r Introduction The advantage

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