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EigCond

# 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 ) , λ ( 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 . Thus we say the relative (and absolute) condition numbers for λ are κ ( λ ) = k A k | λ | 1 | y * x | ( and ν ( λ ) = 1 | y * x | ) . So, if y * x is small, λ is illconditioned. If λ is simple, y * x cannot be zero, but for some matrices it can be very small. Let’s take a diagonalizable matrix under consideration. Suppose X - 1 AX = Λ = diag( λ 1 , . . . , λ n ). The columns of X are (normalized) right eigenvectors of A , while the rows of X - 1 = DY are (un-normalized) left eigenvectors. Let
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