18.03 Class 36
, May 5, 2010
The matrix exponential: initial value problems.
1. Definition of e^{At}
2. Computation of e^{At}
3. Uncoupled example
4. Defective example
5. Exponential law
[1]
Recall from day one:
(a)
x' = rx
with initial condition
x(0) = 1
has solution
x
=
e^{rt} .
x' = rx
with any initial condition has solution
x = e^{rt} x(0) .
Later, we decided to *define*
e^{it}
as the solution to
(b)
x' = ix
with initial condition
x(0) = 1 .
Following Euler, a solution is given by
cos t + i sin t ,
so we found that
e^{it} = cos(t) + i sin(t)
(c)
Now we are studying
u' = A u
.
Let's try to *define*
The solution to
u'
=
Au
with initial condition
u(0)
is
u
=
e^{At}u(0).
(**)
Note that the initial value
u(0)
is a vector, and
u(t)
is a vector
valued function.
So the expression
e^{At}
must denote a matrix, or
rather a matrix valued function.
What could
e^{At}
be?
For a start, what is its first column?
Recall that the first column of any matrix
B
is
the product
B[1;0] ,
and
[ b_1
b_2 ] [1;0] = b_1 , so combining this with (**) we see:
The first column of
e^{At}
is the solution to
u' = Au
with
u(0) = [1;0].
Similarly,
The second column of
e^{At}
is the solution to
u' = Au
with
u(0)= [0;1].
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 Spring '09
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