Embree draft 23 February 2012 18 Nondiagonalizable Matrices The Jordan Form 35

Embree draft 23 february 2012 18 nondiagonalizable

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Embree – draft – 23 February 2012
1.8. Nondiagonalizable Matrices: The Jordan Form 35 Characteristic Polynomial and Minimal Polynomial Suppose the matrix A n × n has the Jordan structure detailed in Theorem 1.7. The characteristic polynomial of A is given by p A ( z ) = p
j =1 ( z λ j ) a j , a degree- n polynomial; the minimal polynomial of A is m A ( z ) = p
j =1 ( z λ j ) d j . Since the algebraic multiplicity can never exceed the index of an eigen- value (the size of the largest Jordan block), we see that m A is a polynomial of degree no greater than n . If A is not derogatory, then p A = m A ; if A is derogatory, then the degree of m A is strictly less than the degree of p A , and m A divides p A . Since ( J j λ j I ) d j = N d j j = 0 , notice that p A ( J j ) = m A ( J j ) = 0 , j = 1 , . . . , p, and so p A ( A ) = m A ( J j ) = 0 . Thus, both the characteristic polynomial and minimal polynomials annihi- late A . In fact, m A is the lowest degree polynomial that annihilates A . The first fact, proved in the 2 × 2 and 3 × 3 case by Cayley [Cay58], is known as the Cayley–Hamilton Theorem Cayley–Hamilton Theorem Theorem 1.8. The characteristic polynomial p A of A n × n annihi- lates A : p A ( A ) = 0 . Embree – draft – 23 February 2012
36 Chapter 1. Basic Spectral Theory 1.9 Analytic Approach to Spectral Theory The approach to the Jordan form in the last section was highly algebraic: Jordan chains were constructed by repeatedly applying A λ I to eigen- vectors of the highest grade, and these were assembled to form a basis for the invariant subspace associated with λ . Once the hard work of construct- ing these Jordan chains is complete, results about the corresponding spec- tral projectors and nilpotents can be proved directly from the fact that V 1 V = I . In this section we briefly mention an entirely di ff erent approach, one based much more on analysis rather than algebra, and for that reason one more readily suitable to infinite dimensional matrices. This approach gives ready formulas for the spectral projectors and nilpotents, but more work would be required to determine the properties of these matrices. To begin, recall that resolvent R ( z ) := ( z I A ) 1 defined in (1.5) on page 10 for all z that are not eigenvalues of A . Recall from Section 1.3 that the resolvent is a rational function of the parameter z . Suppose that A has p distinct eigenvalues, λ 1 , . . . , λ p , and for each of these eigenvalues, let Γ j denote a small circle in the complex plane centered at λ j with radius su ciently small that no other eigenvalue is on or in the interior Γ j . Then we can then define the spectral projector and spectral nilpotent for λ j : P j := 1 2 π i
Γ j R ( z ) d z, D j := 1 2 π i
Γ j ( z λ j ) R ( z ) d z. (1.28) In these definitions the integrals are taken entrywise. For example, the matrix and resolvent A = 1 0 0 0 2 0 1 0 1 , R ( z ) = 1 z 1 0 0 0 1 z 2 0 1 ( z 1) 2 0 1 z 1 with eigenvalues λ 1 = 1 and λ 2 = 2 has spectral projectors P 1 = 1 2 π i
Γ 1 1 z 1 d z 0 0 0
Γ 1 1 z 2 d z 0
Γ 1 1 ( z 1) 2 d z 0
Γ 1 1 z 1 d z = 1 0 0 0 0 0 0 0 1

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