a 2 2 if we omit a piece of a matrix the norm does

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Unformatted text preview: or matrices. We also show that a matrix and its transpose have the same two norm. Fact 2.19 (Two Norm). Let A ∈ Cm×n . Then A∗ A∗ A = A 2, 2 2 = A 2. 2 Proof. The definition of the two norm implies that for some x ∈ Cn with x we have A 2 = Ax 2 . The definition of the vector two norm implies A 2 2 = Ax 2 2 = x ∗ A∗ Ax ≤ x 2 A∗ Ax 2 2 =1 ≤ A∗ A 2 , where the first inequality follows from the Cauchy–Schwarz inequality in Fact 2.8 and the second inequality from the two norm of A∗ A. Hence A 2 ≤ A∗ A 2 . 2 Fact 2.18 implies A∗ A 2 ≤ A∗ 2 A 2 . As a consequence, A 2 2 ≤ A∗ A 2 ≤ A∗ 2 A 2, A 2 ≤ A∗ 2 . Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 36 2. Sensitivity, Errors, and Norms The same reasoning applied to AA∗ gives A∗ Therefore A∗ 2 2 2 ≤ AA∗ =A 2 2 ≤A and A∗ A 2 2 A∗ 2 , A∗ 2 ≤ A 2. = A 2. 2 If we omit a piece of a matrix, the norm does not increase but it can decrease. Fact 2.20 (Norm of a Submatrix). Let A ∈ Cm×n . If B is a submatrix of A, then B p ≤ A p. Exercises (i) Let D ∈ Cn×n be a diagonal matrix with diagonal elements djj . Show that D p = max1≤j ≤n |djj |. (ii) Let A...
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