So, let's read this definition. This
definition first says that we start
with a matrix A, the first thing this
requires is that this matrix is a matrix
that has the same number of rows as the
number of columns, so columns, it has as
many rows as columns, this is square matrix,
so that's the first requirement, thing,
the first thing required in this
definition, and then it says considered a
vector that is not a vector of zeros, so
we're thinking of this being a column
vector and this says that, we're going to
consider this non-trivial vector a
vector doesn't contain all zeros, maybe
some of the entries are zeros but not
all the entries, and this says look, if
the product of the matrix times that
column vector is this product, this is a
vector in CN, right, so here you're
multiplying an N by n, multiplying that
times an N by 1, so that is giving you
already a column that's an N by one
column, a vector in CN, so if that
multiplication ax is a scalar multiple
of X, we would say that that vector is an
eigenvector for some scalar lambda, this,
this here, this lambda is some scalar for
which this equation is true, and here you,
I want you to, I want to emphasize that
what we have on the left hand side is a
square matrix multiplying a column
vector and what we have on the right
hand side is a scalar multiplying a
column vector so this product on the
left hand side is a column and this
product on the right hand side is also a
column, right, so we would say that this
is an equation but it definitely is a
vector equation, right, so it's not a
matrix equation, it's not a scalar equation,
instead is a vector equation. That means
that there are two columns
with n entries, there are two objects
that are n by 1 and those two objects
are of equal length and they match each
other and so that would be the idea that
this matrix product, matrix vector
product equals this scalar column vector
product, okay, and so that equality in
words says well, turns out that that product Ax
is a scalar multiple of the non-trivial
vector x, so this is the definition
of that such scalar, this is for some
scalar, with that such scalar with this scalar here is what we would call
eigenvalue, and the non trivial
vector for which this vector equation is
true is going to be the eigenvector,
so in words we would write the scalar
alpha, the scalar, excuse me,
lambda is called an eigenvalue of the
matrix A and x is an eigenvector
corresponding to lambda so that it will
be an idea, a notion of how in a vector for
which like this vector equation is true
given a particular value lambda and so
if that is the case we would say that
vector, that eigenvector could response
to that scalar that is called
an eigenvalue, so in a way is like they
come as a group of two, as a team, and
however, that team has one scalar and one
vector and the scalar is called maybe
should be called eigenscalar but it's
called eigenvalue, and that vector that is
specific to that a eigenvalue is called, is
called eigenvector.
