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, x 2 ≤ 1}. Problem 2 (30%) Let A ∈ Rm×n . The following function A a,b = sup{ Ax b | x a ≤ 1}, is called an induced norm of A. Here, · a , · b are some given vector norms, and sup denotes the supremum (or simply speaking but somehow inaccurately speaking, maximum). (a) Verify that A (b) Suppose · a a,b = is a norm. · b = · 1. Show that n A a,b |aij |. = max 1≤j≤n 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 m X F n |xij |2 . = i=1 j=1 Verify that X 2 F = tr(XH X). (b) Let A = BH B, B ∈ Cm×n . Verify that zH Az ≥ 0, for all z ∈ Cn . 1

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