18.03 Class 32, May 1
Eigenvalues and eigenvectors
[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 = y = 0. 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 in 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:
combination of the entries in A
"determinant" of the matrix:
c/a = d/b , or ad  bc = 0. This
is so important it's called the
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 = [1 2 ; 2 1] .
The "Linear Phase Portraits: Matrix Entry" Mathlet shows that some
trajectories seem to be along straight lines. Let's find them first.
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 Winter '08
 Staff
 Linear Algebra, Algebra, Eigenvectors, Vectors, Det, lambda

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