2 a 1 and 2a 1a but for a general matrix we cannot

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Unformatted text preview: | + | sin θ|, | − sin θ| + | cos θ|} = | cos θ| + | sin θ|, Then = [x2 cos2 θ + x2 sin2 θ − 2x1x2 cos θ sin θ 1 2 Thus A 1 −1 and = [x2 + x2 ]1/2 = x 1 2 terms of θ). 2 ≤ A 1, and κ2(A) ≤ κ1(A), but for a general matrix, we cannot tell which norm or condition number is larger. 2 Ax 2 = max 1 = 1. x=0 x2 = 1. Then κ2(A) = 1. A geometric way of viewing this: First, note that the condition number of a matrix denotes the ratio of the maximal stretching over the Tut5 – Norms, condition numbers c C. Christara, 2012-13 5 Tut5 – Norms, condition numbers 7 c C. Christara, 2012-13 Tut5 – Norms, condition numbers 8 c C. Christara, 2012-13 minimal stretching (or maximal shrinking) that the matrix gives rise to, when applied to any non-zero vector: κa (A) = ||A||a||A−1||a = = max x=0 Ax maxx=0 ||||x||||a ||A−1x||a ||Ax||a ||y ||a ||Ax||a a max = max max = ||Ay ||a x=0 x=0 ||x||a y =0 ||Ay ||a ||x||a ||x||a miny =0 ||y ||a Then, notice that A represents a counter-clockwise rotation of x by θ radians. We can see that the Euclidian norm (length) of x does not change if A is applied to it. Since A produces neither stretching nor shrinking when applied to any vector, it has condition number 1 with respect to Euclidian norm. Another way of viewing this: (Ax)T Ax ||Ax||2 It is easy to see that A is orthogonal, i.e., AT A = I. Then ||A||2 = max = max √ = x=0 ||x||2 x=0 xT x √ √ xT AT Ax xT x max √ = max √ = 1. Also, since A is square (and orthogonal), its inverse is its transx=0 x=0 xT x xT x pose, i.e., A−1 = AT . So we have AAT = I, i.e., AT is also orthogonal (and square). Thus ||AT ||2 = 1, and thus κ2(A) = 1. Note: For the above matrix A, it is easy to find κ2(A). For arbitrary matrices, it is not straightforward. Even if we have the inverse explicitly, it is not always easy to calculate A Tut5 – Norms, condition numbers 6 2 and A−1 2. c C. Christara, 2012-13 Q UESTION 7 Let A = ANSWER: It is easy to see that A−1 = A ∞ = max{8, 2} = 8, A−1 κ∞(A) = A...
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