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18.03 Class 33
, Apr 28, 2010
Eigenvalues and eigenvectors
1. Linear algebra
2. Ray solutions
3. Eigenvalues
4. Eigenvectors
5. Initial values
[1]
Prologue on Linear Algebra.
Recall
[a b ; c d] [x ; y] = x[a ; c] + y[b ; d] :
A matrix times a column vector is the linear combination of
the columns of the matrix weighted by the entries in the column vector.
When is this product zero?
One way is for
x = 0 = y.
If
[a ; c]
and
[b ; d]
point in
different directions, this is the ONLY way. But if they lie along a
single line, we can find
x
and
y
so that the sum cancels.
Write
A = [a b ; c d]
and
u = [x ; y] , so we have been thinking
about
A u = 0
as an equation for
u . It always has the "trivial"
solution
u = 0 = [0 ; 0] :
0
is a linear combination of the two
columns in a "trivial" way, with
0
coefficients, and we are asking
when it is a linear combination of them in a different, "nontrivial" way.
We get a nonzero solution
[x ; y]
exactly when the slopes of the vectors
[a ; c]
and
[b ; d]
coincide:
c/a = d/b ,
or
ad  bc = 0.
This
combination of the entries in
A
is so important that it's called the
"determinant" of the matrix:
det(A) = ad  bc
We have found:
Theorem:
Au = 0
has a nontrivial solution exactly when
det A = 0 .
If A is a larger *square* matrix the same theorem still holds, with
the appropriate definition of the number
det A .
[2]
Solve
u' = Au : for example with
A = [2 1 ; 4 3] .
The Mathlet "Linear Phase Portraits: Matrix entry" shows that some
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 Spring '09
 vogan
 Linear Algebra, Algebra, Eigenvectors, Vectors

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