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 ndimensional 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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 Spring '09
 Linear Algebra, Vector Space, lim, Norm, Euclidean space

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