Lecture #4 Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from September 6, 2012 taken by Nathaniel Hammen Last Time (9/4/12) Diagonalization: conditions for diagonalization Eigenvalue Multiplicity: algebraic and geometric multiplicity 1 Further Review 1.1 Warm-up questions 1.1.1 Question. If A M n has only one eigenvalue λ (of multiplicity n )andisd iagona l izab le , what is A ? Answer. Because A is diagonalizable, there exists an invertible S M n such that S 1 AS is diagonal. In fact, because all eigenvalues are λ ,wehave A = S ( S 1 AS ) S 1 = SλIS 1 = λSS 1 = λI 1.1.2 Question. Let A M n and f ( t )= det ( I + tA ) .Wha ti s f ° ( t ) in terms of A ? Answer. The i, j th entry of ( I + tA ) is δ i,j + ta i,j ,so f ( t det ( I + tA ° σ S n sgn ( σ ) n ± j =1 ( δ σ ( j ) ,j + ta σ ( j ) ,j ) = ° σ S n sgn ( σ ) ² n ± j =1 δ σ ( j ) ,j + t n ° j =1 a σ ( j ) ,j ± i ± = j δ σ ( i ) ,i + o ( t 2 ) ³ where S n is the set of permutations of n elements and sgn ( σ ) is +1 if σ is an even permutation and -1 if σ is an odd permutation. DiFerentiating f ( t ) gives f ° ( t ° σ S n sgn ( σ ) ² n ° j =1 a σ ( j ) ,j ± i ± = j δ σ ( i ) ,i + o ( t ) ³ = n ° j =1 a j,j + o ( t tr ( A )+ o ( t ) 1
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This last equality holds because ° i ° = j δ σ ( i ) ,i is only nonzero when σ ( i )= i for all i ° = j ,wh ich only occurs when σ is the identity. When we plug 0 into the equation, we get that f ± (0) = tr ( A ) . 1.4 Similarity (cont’d) 1.4.27 Theorem. Amatr ix A M n is diagonalizable if and only if the geometric and algebraic multiplicities of each eigenvalue are equal. Proof. We begin by noting that if λ j and λ k are eigenvalues of A with λ j ° = λ k ,th enth e i r eigenspaces intersect trivially; that is E λ j E λ k = { 0 } .T oshowth i s ,l e t x E λ j E λ k .Th en λ j x = Ax = λ k x ichimp l ie s x =0 because λ j ° = λ k h u si f { v 1 ,v 2 ,...,v m j } is a basis for E λ j and { u 1 ,u 2 ,...,u m k } is a basis for E λ k ,then { v 1 2 m j }∪{ u 1 2 m k } is a linearly independent set, and forms a basis for E λ j + E λ k .Induc t ive lyi te ra t ingth i s ,w eob ta ina basis for E λ 1 +
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Lecture #4 Notes - Matrix Theory Math6304 Lecture Notes...

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