hw4 - ECE210A October 18, 2010 Home work 4 Due date:...

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Unformatted text preview: ECE210A October 18, 2010 Home work 4 Due date: October 25, 2010 1. Reading assignment. Read chapters 3 and 4 of the class notes. 2. Show that for p 1, and p−1 + q −1 = 1, x p = max 0 y y Tx . yq See the class notes for hints. Holder’s inequality is useful. 3. Let A−1 E < 1, for some induced matrix norm. Show that (A + E )−1 exists and that (A + E )−1 − A −1 A −1 E 1 A 1 − A −1 A −1 A E . See the class notes. 4. Show that A 2 A 1 A ∞. [This is quite difficult to prove (as far as I know). See the 2 class notes on how to proceed.] 5. We say that a sequence x(1), x(2), x(3), vector x if , of n-dimensional vectors converges to the lim x(k) − x = 0. k ↑∞ Show that x(k) converges to x, if and only if, for i = 1, lim |x(k) − xi | = 0, i k ↑∞ , n. 6. Let . b e a vector norm on Rm and assume A ∈ Rm × n. Show that if rank(A) = n, then x A ≡ A x is a vector norm on Rn. 7. Prove or disprove that for v ∈ Rn v 1 v √ 1+ n v 2. 2 2 ∞ 8. Let B be a submatrix of A. Show that B n 9. Show that if s ∈ R and s 0, and E ∈ R E I− s sT sTs A p p, for 1 ∞. p n ×n , then 2 =E F 2 F − Es 2 2 . sT s 10. Suppose u ∈ Rm and v ∈ Rn. Show that if E = u v T , then E E ∞ = u ∞ v 1. 1 F =E 2 =u 2 v 2, and ...
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