EigCond - Sensitivity of Simple Eigenvalues Let A Cnn . We...

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Sensitivity of Simple Eigenvalues Let A C n × n . We would like to know how small perturbations in A change its eigenvalues. Of course ∂λ k /∂a ij measures just this sensitivity, but it isn’t practical to compute these n 3 quantities. Suppose Ax = λx , y * A = λy * , and k x k 2 = 1 = k y k 2 (so x and y * are respectively right and left eigenvectors associated with a simple (not multple) eigenvalue λ of A ). Remember, y * = ¯ y t is a conjugate transpose. Consider the perturbed eigenpair x ( t ) ( t ) of the matrix A + tE : ( A + tE ) x ( t ) = λ ( t ) x ( t ) , where x (0) = x , λ (0) = λ , and k x ( t ) k 2 = 1. Differentiating wrt t gives: ( A + tE ) ˙ x ( t ) + Ex ( t ) = λ ˙ x ( t ) + ˙ λx ( t ) . Premultiplying by y * gives ˙ λ (0) = y * Ex/ ( y * x ), and for λ 6 = 0, Taylor’s theorem says | λ ( t ) - λ λ | = | t ˙ λ λ + O( t 2 ) | | ty * Ex λy * x | k A k | λy * x | k tE k k A k .
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

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