okay in this video we're going to do a
little introduction to eigenvalues and
eigenvectors and all i'm going to do in
this video is just give some definitions
a little notation do one very short
example and that'll be it and the next
video will actually look at how we go
about finding eigenvalues and
eigenvectors
okay so definition here let a be an N by
n matrix and we say that a scalar lambda
is called an eigenvalue of a if there's
a nonzero vector X so that when we take
our matrix a and multiply it by X we get
the exact same thing as if we take our
scalar lambda and multiply it by X such
a vector X is called an eigenvector of a
corresponding to lambda okay so you'll
have different eigenvectors that go with
different eigenvalues and again i ghen
values and eigenvectors are super super
super useful super important have tons
and tons of applications and we'll talk
about some of those down the road as
well so here's going to be our quick
little example here again nothing crazy
we're going to show that our vector X
that has entries 2 & 1 we're going to
show that that's an eigenvector of the
matrix a that has entries 3 2 3 and
negative 2 corresponding to the value of
lambda equals 4 okay I'm not saying that
this matrix a actually has other eigen
values we'll talk about that as well
this matrix also has other eigenvectors
and we'll talk about that in just a
second okay so really all we have to do
is we want to show is the equation a
times X does that equal lambda times X
so not a lot to do here other than just
a bit of arithmetic so our matrix a has
entries 3 2 3 and negative 2 again our
vector X has entries 2 and 1 and I'm
asking myself well if is that going to
be equal
to the right side when we take lambda
which is 4 and again multiply that by X
well all we have to do is just a bit of
arithmetic so on the left side we'll do
our matrix multiplication 3 x 2 plus 2 x
1 that'll be our first row and then our
second row will have 3 x 2 plus then
we'll take negative 2 x 1 well on the
right side we just multiply our entries
2 and 1 by 4 well 4 times 2 is 8 4 times
1 is 4 well on the left side 3 times 2
is 6 plus 2 times 1 will give us 8 3
times 2 is 6 we'll have plus negative 2
so 6 minus 2 will give us 4 hey that
certainly does give us a quality so yes
in fact we would say that the vector
with entries 2 & 1 is an eigen vector of
the matrix a corresponding to the eigen
value of 4 okay so one other little
remark here if lambda is an eigen value
of a so we had 4 was an eigen value of
our matrix a and X is an eigen vector
belonging to lambda so again our
eigenvector was the inch with the vector
with entries 2 and 1 that corresponded
to the eigen value of 4 since any
nonzero multiple of X will be an eigen
vector as well okay so it turns out that
since two and one was an eigen vector we
can take any nonzero multiples so 4 + 2
xx + 10 200 and 100 all of those values
would have also been I can vectors that
correspond to the eigen value of 4 and
again we can show this quickly
we're already assuming that a times x
equals lambda times X ok well to show it
in this case
if we take our our eigen vector X and
multiply it by some constant alpha well
we can always simply rewrite this since
alpha is a scalar we can pull that out
front and have alpha times a times X but
again by assumption a times X that
equals lambda times X and again since
alpha is just a scalar we can put that
back inside and we'll have that lambda
equals alpha times X and now again we've
got our desired result
so all right again nothing nothing too
heavy in this example in this video and
the next one again we'll start actually
looking at finding eigenvalues and
eigenvectors in general because that's
probably more to the point and something
that you'll certainly see certainly be
asked to do in a linear algebra class