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ENGG 5781: Matrix Analysis and Computations
2014–15 Second Term
Assignment 1 Instructor: Wing-Kin Ma
Due: 5:30pm, January 28, 2015
Problem 1 (30%) In each of the subsets below, is it a subspace, or not? Provide your answer with a concise verification. (a) S = S1 ∪ S2 , where S1 , S2 ⊆ Rm are subspaces. (b) S = S1 ∩ S2 , where S1 , S2 ⊆ Rm are subspaces. (c) S = {z ∈ Rn | z = [ y1 , y2 , . . . , yn ]T , y ∈ S1 }, where S1 ⊆ Rm is a subspace and n < m. (d) direct sum of two subspaces; i.e., S = {y ∈ Rm | y = y1 + y2 , y1 ∈ S1 , y2 ∈ S2 }, where S1 , S2 ⊆ Rm are subspaces. (e) S = {y ∈ Rm | yT x = 0 ∀x ∈ B}, where B = {x ∈ Rm | xm = 0, kxk2 ≤ 1}. Problem 2 (30%) Let A ∈ Rm×n . The following function kAka,b = sup{kAxkb | kxka ≤ 1}, is called an induced norm of A. Here, k · ka , k · kb are some given vector norms, and sup denotes the supremum (or simply speaking but somehow inaccurately speaking, maximum). (a) Verify that kAka,b is a norm. (b) Suppose k · ka = k · kb = k · k1 . Show that kAka,b = max
1≤j≤n
n X
|aij |.
i=1
Problem 3 (20%) Let A ∈ Rm×k , B ∈ Rk×n . (a) Prove, in a rigorous way and solely by the definition of rank, that rank(AB) ≤ rank(B). (b) Concisely argue how you may use the result in (a) to further conclude rank(AB) ≤ min{rank(A), rank(B)}. Problem 4 (20%) (a) Let X ∈ Cm×n , and define v uX n um X kXkF = t |xij |2 . i=1 j=1
Verify that
kXk2F
=
tr(XH X).
(b) Let A = BH B, B ∈ Cm×n . Verify that zH Az ≥ 0, for all z ∈ Cn .
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