3 and 9. The operator norm on matrices. If V and W are

HW 3

Cookbook problems. Solve:

1) Cookbook-I:13 and22.

2) Cookbook-II:3and9.

The operator norm on matrices. IfV andW are vector spaces over a fieldF, we write Hom_F(V, W) for the vector space of linear transformations fromV toW. Recall from §§2.1 of Lecture 5 in ANA2 that the vector space of linear operators from Rⁿ to Rⁿ has a norm. The norm was denoted there byk k_L. However, in this course,Lwill usually be used for Lipschitz constants and hence we will use the symbolk k_◦ for the operator norm. Recall also that we havekABk◦ ≤ kAk◦kBk◦

for A ∈ HomR(R^k,R^m) and B ∈ HomR(Rⁿ,R^k) (see 7 of §1 of Lecture 7 in ANA2).

In this HW, if we say I is an interval, then we allow I to be open, closed, or half-open, but insist that it have a non-empty interior. This means single points are not intervals for us.

We often writeM_m,n(F) for the space ofm×nmatrices with entries in a field F. ClearlyM_m,n(F) can be identified with Hom_F(Fⁿ, F^m), and we will often do so implictly. Thus it makes sense to talk about the operator norm on Mm,n(R), and we will denote this also by k k_◦. Since Mm,n(R) (which we identify with HomR(Rⁿ,R^m)) is finite dimensional, all norms on it are equivalent. In particular, a map from an intervalI to (Mm,n(R),k k_◦), sayA:I→Mm,n(R), is continuous if and only if each of the entries of the matrix of functionsAis continuous.¹ Lett◦

be a point inI. We sayA: →M_m,n(R) isdifferentiable att◦ if:

(∗) lim

h→0

1 h

A(t◦ +h)−A(t◦)

exists. The above limit is the appropriate one-sided limit in the event our interval is closed or half open, andt◦ is a boundary point ofI. These limits are of course taken with respect to k k_◦. However, as we just argued, any norm can be used, since all norms on Mm,n(R) are equivalent. If A is differentiable at t◦ then the limit in (∗) is called thederivative of Aat t◦ and denoted by any of the familiar symbols ^dA_dt|_t=t◦, .

A(t◦), ^dA_dt(t◦),A⁰(t◦) etc. Ais said to bedifferentiable on I if it is differentiable at every point ofI. In such a case, the functiont7→ .

A(t) is called the derivative ofAonI, and is denoted by the symbols .

A, ^dA_dt,A⁰, and sometimes as ^dA(t)_dt or _dt^dA(t).

1A functionA:S→Mm,n(R), whereSis some non-empty set, is the same as anm×nmatrix of functionsA= (aij) withaij:S→R, 1≤i≤m, 1≤j≤n.

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3) LetIbe an interval inRand suppose

x(t) =. A(t)x(t) (t∈I)

is a linear first order differential equation withAann×nmatrix of continuous functions on an I. Supposey₁(t), . . . ,y_n(t) are nsolutions of this differential equation onI. Let

W =W(y₁, . . . ,y_n) :I−→R be the function given by

W(t) = det [y(t), . . . ,y_n(t)] (t∈I),

where [y(t), . . . ,y_n(t)] is regarded an n×n matrix whose i^th column isy_i(t), i= 1, . . . , n. Show that either W is identically zero onI or it is nowhere van- ishing onI.

4) Let I be an interval in R. Suppose A: I → M_m,k(R) and B: I → M_k,n(R) are differentiable on I. Show that t 7→A(t)B(t) gives us a differentiable map AB:I→M_m,n(R) and that

d dt

A(t)B(t)

= .

A(t)B(t) +A(t).

B(t) (t∈I).

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