Linear Algebra, Infinite Dimensional Spaces, and MAPLE

83 Pages·1997·0.272 MB·English

by  Herod J.V.

Math 6580
One must be sure that one has enabled science to make a great advance if one
is to burden it with many new terms and require that readers follow research
that offers them so much that is strange.
A. L. Cauchy
Linear Algebra, Infinite Dimensional Spaces,
and MAPLE
PREFACE
These notes evolved in the process of teaching a beginning graduate
course in Hilbert Spaces. The first edition was simply personal, handwritten
notes prepared for lecture. A copy was typed and given to the students, for it
seemed appropriate that they should have a statement of theorems,
examples, and assignments. With each teaching of the course, the notes grew.
A text for the course has always been announced, but purchase of the
text has been optional. A text provides additional reading, alternate perspec-
tives, and a source of exercises. Whatever text was used, it was chosen with
the intent of the course in mind.
This one quarter course was designed for science and engineering
students. Often, as graduate students in the sciences and engineering
mature, they discover that the literature they are reading makes references
to function spaces, to notions of convergence, and to approximations in
unexpected norms.
The problem they face is how to learn about these ideas without
investing years of work which might carry them far from their science and
engineering studies. This course is an attempt to provide a way to understand
the ideas without the students already having the mathematical maturity
that a good undergraduate analysis course could provide. An advantage for
the instructor of this course is that the students understand that they need to
know this subject.
The course does not develop the integration theory and notion of a
measure that one should properly understand in order to discuss L2[0,1]. Yet,
these note suggest examples in that space. While this causes some students
to feel uneasy with their unsophisticated background, most have enough
intuition about integration that they understand the nature of the examples.
The success of the course is indicated from the fact that science and
engineering students often choose to come back the next term for a course in
real variables and in functional analysis. The course has been provocative.
Even those that do not continue with more graduate mathematics seem to
feel it serves them well and provides an opening for future conversations in
dynamics, control, and analysis.
In this most current revision of the notes, syntax for MAPLE has been
added. Many sophomores leave the calculus thinking that the computer
algebra systems are teaching tools because of the system's abilities to graph,
to take derivatives, and to solve text-book differential equations. We hope,
before they graduate, students will find that these systems really are a "way
for doing mathematics." They provide a tool for arithmetic, for solving
equations, for numerical simulations, for drawing graphs, and more. It's all in
one program! These computer algebra systems will move up the curriculum.
In choosing MAPLE, I asked for a computer algebra system which is
inexpensive for the students, which runs on a small platform, and which has
an intuitive syntax. Also, MATLAB will run Maple syntax.
The syntax given in these notes is not always the most efficient one for
writing the code. I believe that it has the advantage of being intuitive. One
hopes the student will see the code and say, "I understand that. I can do it,
too." Better yet, the student may say, "I can write better code!"
It is most important to remember that these notes are about linear
operators on Hilbert Spaces. The notes, the syntax, and the presentation
should not interfere with that subject.
The idea in these notes that has served the students best is the presen-
tation of a paradigm for a linear operator on a Hilbert Space. That paradigm
is rich enough to include all compact operators. For the student who is in the
process of studying for various exams, it provides a system for thinking of a
linear function that might have a particular property. For the student who
continues in the study of graduate mathematics, it is a proper place to step
off into a study of the spectral representations for linear operators.
To my colleagues, Neil Calkin and Eric Bussian, I say: The greatest
complement you can give a writer is to read what he has written. To my
students: These notes are better because they read them and gently
suggested changes.
I am grateful to the students through the years who have provided
suggestions, corrections, examples, and who have demanded answers for the
assignments. In time.... In time....
James V. Herod
School of Mathematics
Georgia Tech
Atlanta 30332-0160
Summer, 1997
Table of Contents
Section Title Page
1 A Decomposition for Matrices 1
2 Exp(tA) 5
3 Self Adjoint Transformations in Inner-Product Spaces 7
4 The Gerschgorin Circle Theorem 11
5 Convergence 14
6 Orthogonality and Closest Point Projections 18
7 Orthogonal, Nonexpansive, & Self-Adjoint Projections 23
8 Orthonormal Vectors 27
9 The Finite Dimensional Paradigm 32
10 Bounded, Linear Maps from E to CI 35
11 Applications to Differential and Integral Equations 40
12 The Simple Paradigm for Linear Mappings from E to E 43
13 Adjoint Operators 45
14 Compact Sets 47
15 Compact Operators 50
16 The Space of Bounded Linear Transformations 53
17 The Eigenvalue Problem 57
18 Normal Operators and The More General Paradigm 61
19 Compact Operators and Orthonormal Families 64
20 The Most General Paradigm:
A Characterization of Compact Operators 67
21 The Fredholm Alternative Theorems 69
22 Closed Operators 72
23 Deficiency Index 74
24 An Application: A Problem in Control 77
26 An Application: Minimization with Constraints ??
25 A Reproducing Kernel Hilbert Space ??
1
Linear Algebra, Infinite Dimensional Spaces, and MAPLE
This course will be chiefly concerned with linear operators on Hilbert
Spaces. We intend to present a model, a paradigm, for how a linear
transformation on an inner-product space might be constructed. This
paradigm will not model all such linear mappings. To model them all
would require an understanding of measures and integration beyond what
a beginning science or engineering student might know.
To set the pattern for this paradigm, we first recall some linear
algebra. We recall, review, and re-examine the finite dimensional setting.
In that setting, there is the Jordan Canonical Form. We present this
decomposition for matrices in an alternate view from the traditional one.
The advantage to the representation presented here is conceptual. It sets a
pattern in concepts, instead of in "form." We think of projections and
eigenvalues. And, we must turn to nilpotent matrices when projections and
eigenvalues are not enough.
This is how we begin.
Section 1: A Decomposition for Matrices
Definition A projection is a transformation P from E fi E such that P2 = P.
Note that some texts also require that P should be non-expansive in the
sense that |Px| £ |x|. An example of a projection that is not non-
expansive is P(x,y) = (x+y,0)
Definition A nilpotent transformation is a linear transformation N from E
fi E for which there is an integer k such that Nk = 0.
(cid:230) 3 0-2(cid:246) (cid:230) -1 1 1(cid:246)
Examples (cid:231) -1 1 1(cid:247) is a projection from R3 to R3 and (cid:231) 0 0 0(cid:247) is a nilpotent
Ł 3 0-2ł Ł -1 1 1ł
function from R3 to R3.
Theorem 1(Spectral Resolution for A) If A is a linear function from Rn to
k k k
Rn then there are sequences {l } , {P } , and {N } such that each l is
i i=1 i i=1 i i=1 i
a number and
(a) P is a projection, (b) N is nilpotent, (c)P P = 0 if i „ j
i i i j
k
(d) N P = 0 if i „ j, (e) N P = N (f) I = (cid:229) P ,
i j i i i i
1
k
and (g) A = (cid:229) [l P + N ]
i i i
i
2
Outline of Proof: We assume the Cayley-Hamilton Theorem which states
that if A in an nxn matrix and D(l ) is the polynomial in l defined by D(l ) =
det(l I-A), then the polynomial in A defined by making the substitution A =
l satisfies D(A) = 0.
To construct the proof for the theorem, first factor D(l ) as
k
D(l ) = (cid:213) (l -l )mp,
p
p=1
where the l p's are the zero's of D with multiplicity m . Now form a partial
p
fraction decomposition of D(l ): Let a , a , ... a be functions such that
1 2 k
1 (cid:229)k ap(l )
= .
D(l ) p=1 (l -l p)mp
Two Examples
(cid:230) 1 0 0(cid:246)
(cid:230) -1-2(cid:246)
A1 = Ł 3 4ł and A2 = (cid:231) 0 2 1(cid:247)
Ł 0 0 2ł
D (l ) = (l -1)(l -2) and D (l ) = (l -1)(l -2)2
1 2
1 -1 1
= +
(l -1)(l -2) l -1 l -2
1 1 -l +3
= +
(l -1)(l -2)2 l -1 (l -2)2

If q (l ) = (cid:213) (l -l )mi then
p i
i„ p
k k
1 = (cid:229) a (l ) q (l ) so that I = (cid:229) a (A) q (A)
p p p p
p=1 p=1
Two Examples Continued
1 = -1(cid:215) (l -2) + (l -1) = (2-l ) + (l -1)
1 = (l -2)2 + (-l +3)(l -1)
(cid:230) 100(cid:246) (cid:230) 000(cid:246)
(cid:230) 3 2(cid:246) (cid:230) -2-2(cid:246)
I = Ł -3-2ł + Ł 3 3ł and I = (cid:231) 000(cid:247) + (cid:231) 010(cid:247) .
Ł 000ł Ł 001ł
Claim 1: Using the Caley-Hamilton Theorem if P = a (A) q (A) then P P =
j j j i j
0.
3
k
Claim 2: P is a projection since P = P (cid:215) I = (cid:229) P P = P 2.
j j j j i j
i=1
Claim 3: By the Caley-Hamilton Theorem, if
N = a (A)q (A)(A-l I) = P (A-l I)
i i i i i i
m
then N i = 0.
i
Claim 4: N P = N and N P = 0.
i i i i j
To see this note that N P = P N = P P (A-l I) = 0
i j j i j i i
and N P = P P (A-l I) = N .
i i i i i i

(cid:229)
Finally, since I = P then
i
i

(cid:229)
A = P A
i
i

= (cid:229) P (l I + A - l I)
i i i
i

= (cid:229) [l P + N ].
i i i
i
Two Examples Finished
(cid:230) -1-2(cid:246) (cid:230) 3 2(cid:246) (cid:230) -2-2(cid:246)
Ł 3 4ł = 1Ł -3-2ł +2Ł 3 3ł and
(cid:230) 100(cid:246) (cid:230) 100(cid:246) (cid:230) 000(cid:246) (cid:230) 000(cid:246)
(cid:231) 021(cid:247) = 1(cid:231) 000(cid:247) + 2(cid:231) 010(cid:247) + (cid:231) 001(cid:247) .
Ł 002ł Ł 000ł Ł 001ł Ł 000ł
Remarks
k
(1) The sequence {l } is the sequence of eigenvalues. If x is in Rn and
i i=1
m - 1
v = P N i (x),
i i i
then v is an eigenvector for A in the sense that
i
l v = A v .
i i i
(2) For the nilpotent part
m £ n.
i
4
In fact, N n must be zero for each i.
i
Assignment Get the spectral resolution (or Jordan Canonical Form) for the
matrices:
(cid:230) -5 1 3(cid:246)
(cid:230) 0 1(cid:246) (cid:230) -3 2(cid:246) (cid:230) 0-1(cid:246)
Ł 1 0ł , Ł 1-2ł , Ł 1 2ł , and (cid:231) 1-2-1(cid:247) .
Ł -4 1 2ł
5
Section 2: Exp(tA)
Often in differential equations -- both in ordinary and partial
differential equations -- one can conceive of the problem as having this form
Z¢ = AZ , Z(0) = c.
If one can make sense of exp(tA), then surely this should be the solution to a
differential equation of the form suggested. Finite dimensional linear
systems beg for such an understanding. More importantly, this
understanding gives direction for the analysis of stability questions in
linear, and nonlinear differential equations.
Here is a review of the linear, finite dimensional case.
Theorem 2 If P is a projection, t is a number, and x is in {E, <(cid:215) ,(cid:215) >}, Then
the sequence
n n
(cid:229) tiPi (cid:229) ti
S(n) = (x) = P(x)
i! i!
i=1 i=1
converges in E.
Recall that whatever norm is used for Rn, if A is an nxn matrix, then there
is a number B such that if x is in Rn, then |Ax| £ B|x|. Moreover, the
least such B is denoted ||A||.
¥
(cid:229) tiAi
Definition exp(tA) = .
i!
i=0
Corollary 3 If P is a projection, exp(tP)(x) = (1-P)(x) + et P(x).
Corollary 4 Suppose that P and Q are projections, PQ = QP = 0, and if t is a
number. Then
exp(tP+tQ) = et P + et Q + (1-P-Q).
Suggestion of Proof. With the suppositions for P and Q, P+Q is also a
projection. Thus, the previous result applies.
m-1
(cid:229) tiNi
Observation If N is nilpotent of order m then exp(tN) = .
i!
i=0
Theorem 5 If A is a linear transformation from Rnfi Rn and
k
A = (cid:229) [l P + N ]
i i i
i
is as in Theorem 1, then
6
exp(tA) = (cid:229)k exp(l t)P ( m(cid:229) i- 1 tjNij ).
i i j!
1 j=0
Suggestion of Proof. Suppose that l is a number. Then
exp(tA) = exp(l t) exp(t[A-l I]).
Suppose that A = S [l P +N ].
i i i i
¥
Then P exp(tA) = el itP (cid:229) tp (A-l I)p
i i p! i
p=0
¥
= el itPi (cid:229) ptp! Pi(cid:215)(A-l iI)p since Pi = Pi2.
p=0
¥
l t (cid:229) tp p
= e i P N .
i p! i
p=0
Thus, exp(tA) = 1(cid:215) exp(tA)
S
= P exp(tA)
i i
m -1
S l t (cid:229) i tp p
= e i P N .
i i p! i
p=0
Assignment
Solve Z¢ = AZ , Z(0) = c, where A is any of the matrices in the previous
assignments.