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Title: Diagonalization and differential equations
Description: Linear algebra course
Description: Linear algebra course
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Diagonalization and Differential
Equations
6
...
The ideas are not used elsewhere in this book
...
At a
initial time, t
=
0 (in hours), the concentration of salt in tank Y is different from the
concentration in tank Z
...
The two tanks are joined by pipes; through one
pipe, solution is pumped from Y to Z at a rate of 20 L/h; through the other, solution is
pumped from Z to Y at the same rate
...
in each tank at time
y(t)
(z/1000)
(20)(z/1000)
dy -0
...
02z
...
(20)(y/1000)
t, z(t)
t (y/1000)
and let
Then the concentration in Y at time
be the
is
kg/L
...
Then for tank Y, salt is fl
...
Since the rate of change is measured by the derivative, we have
By consideration of Z, we get a second differential equation, so
+
=
are the solutions of the system of linear ordinary differential equations:
and
dy -0
...
02z
dtdz
z
...
02y
-0
dt
d [y] [-0
...
02] [y]
dt z 0
...
02 z
+
=
=
...
...
'
t
1s
convernent
to
rewnte
t
1s
system
m the form
h
I
=
·
How can we solve this system? Well, it might be easier if we could change vari
2
2
[-0
...
0022 -0
...
02]
...
04,
[11 -11]
ables so that the
A
=
x
matrix is diagonalized
...
ith p
'W
=
=
l
2
[
4
...
...
Introduce new coordinates
tion
...
0
by the change of coordinates equation, as in Sec-
[�]
d [y*] - [y*]
dt z* z*
Substitute this for
-P
on both sides of the system to obtain
AP
Since the entries in Pare constants, it is easy to check that
d
dt
[yz**] - dtd [yz**]
-P-
-P
Multiply both sides of the system of equations (on the left) by p-i
...
04z*
dt
dt
-
=
and
-
=
These equations are "decoupled," and we can easily solve each of them by using simple
one-variable calculus
...
So, from
b is a constant
...
The only
kx are exponentials of the form
-0
...
To determine the constants
the amounts y(O) and z(O) at the initial time t
all t
...
04t
+ be-0
...
This is the general
and b, we would need to know
0
...
A Practical Solution Procedure
The usual solution procedure takes advantage of the understanding obtained from this
diagonalization argument, but it takes a major shortcut
...
ce,i1
[:]
...
We find the two eigenvalues A
...
2 and the corresponding
eigenvectors v1 and v2, as above
...
This matches the general solution we found above
...
Many of these systems are much larger than the example we
considered
...
PROBLEMS 6
...
(a)
:t [�]
:!_
[y] [
dt z
=
0
...
1
0
...
4
] [y]
z
[! -�J [�]
=
Homework Problems
Bl Find the general solution of each of the following
(b)
systems of linear differential equations
...
(b) Is there any case where you can tell from the
Explain the connection between the statement that
eigenvalues that A is not diagonalizable over JR
...
is an eigenvalue of A with eigenvector v and the
condition det(A
-
/I
...
(Section 6
...
(Section 6
...
1,
...
(a) What conditions on these eigenvalues guaran
tees that A is diagonalizable over JR
...
2)
4 Use the idea suggested in Problem 6
...
D 4 to create
matrices for your classmates to diagonalize
...
2)
5 Suppose that P-1AP = D, where D is a diago
nal matrix with distinct diagonal entries A
...
, /1
...
4)
tion 6
...
If any is
value
...
If it is, give an invertible matrix
Pand a diagonal matrix D such that p-1 AP= D
...
90 0
...
81 0
...
10]
0
...
1 0
...
...
=
(a) What is the dimension of the solution space of
Ax=
o?
Further Problems
Fl (a) Suppose that A and B are square matrices such
that AB
=
BA
...
" That is, if the char
acteristic polynomial is
of A all have algebraic multiplicity 1
...
(b) Give an example to illustrate that the result in
then
part (a) may not be true if A has eigenvalues
with algebraic multiplicity greater than 1
...
tored form
...
By representing 1
with respect to the basis of eigenvectors, show that
•
•
•
(A - ;i1 !)(A - ,12/) ···(A
1
E
(Hint: Write the characteristic polynomial in fac
-
;i,J)x
=
O
for every
JR
...
F4 For an invertible
n x n matrix, use the Cayley
Hamilton Theorem to show that A-1 can be writ
ten as a polynomial of degree less than or equal
-1
to n
in A (that is, a linear combination of
2
11 1
{A - ,
...
Title: Diagonalization and differential equations
Description: Linear algebra course
Description: Linear algebra course