Unformatted text preview: Q T Q = I is an orthogonal matrix and Λ is a diagonal matrix whose diagonal elements are positive and also are the eigenvalues of A . Deﬁne ˜ x = Q T x and ˜ b = Q T b Ax = b and Λ˜ x = ˜ b Given x and ˜ x , deﬁne the sequence x k as the sequence of vectors produced by steepest descent applied to Ax = b and the sequence ˜ x k as the sequence of vectors produced by steepest descent applied to Λ˜ x = ˜ b . Let e k = x k-x and ˜ e k = ˜ x k-˜ x . Show that if ˜ x = Q T x then k e k k 2 = k ˜ e k k 2 , k > 1...
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- Spring '11
- Linear Algebra, Singular value decomposition, Diagonal matrix, Orthogonal matrix, steepest descent, sequence xk