Assignment 1

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 .
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