This lesson we'll answer the question
what are eigenvalues and eigenvectors?
To begin, let's consider matrix A
and vector x and determine matrix A times vector x.
Matrix A times vector x is equal to the vector five, five,
which is a special result because matrix A times vector x
is equal to a scalar multiple of vector x.
In this case, matrix A times vector x
is equal to five times vector x.
When matrix A is an n by n matrix,
lambda is a number or scalar,
and vector x is a non-zero vector
such that matrix A times vector x
equals lambda times vector x.
Then the scalar lambda is an eigenvalue value of matrix A
and vector x is an eigenvector corresponding to lambda.
So again, going back to our example,
since matrix A times vector x
is equal to five times vector x,
lambda equals five is an eigenvalue of matrix A,
and vector x, the vector one, one,
is an eigenvector corresponding to lambda equals five.
Let's take a look at this graphically.
Let's graph vector x as well as matrix A times vector x
on the same coordinate plane.
Graphically we can easily see matrix A times vector x
or the vector five, five is a scalar multiple of vector x.
We can also say that matrix A times vector x
and vector x are colinear with the origin
as well as parallel vectors.
Which will always be the case
when the vector equation matrix A times vector x
equals lambda times vector x is satisfied.
But this is not always the case.
For example, if we take a look at the same matrix A
and now consider the vector negative two, three,
let's calculate matrix A times vector x.
In this case, matrix A times vector x
is the vector five, zero.
We can quickly see vector x and matrix A times vector x
are not colinear or parallel,
which indicates the eigenvector equation is not satisfied.
Let's take a look at two more examples.
Looking at a different matrix A and a different vector x,
again, let's calculate matrix A times vector x,
which is equal to the vector three, three.
And the vector three, three is a scalar multiple
of the vector negative one, negative one,
since negative three times the vector
negative one, negative one equals the vector three, three.
We can see from the graph
the two vectors are colinear with the origin
as well as parallel.
And therefore lambda equals negative three is an eigenvalue
of matrix A and vector x,
the vector negative one, negative one,
is an eigenvector corresponding to
lambda equals negative three.
And for one more example,
let's consider a different matrix A but the same vector x,
the vector negative one, negative one.
And again, calculate matrix A times vector x.
Matrix A times vector x in this case
is the vector two, three.
We can see from the graph
the two vectors are not parallel
or colinear with the origin
since the vector two, three is not a scalar multiple
of the vector negative one, negative one.
Which means the eigenvector equation is not satisfied
with this matrix A and this vector x.
Before we go, let's take a look at one reason
why we care about eigenvalues.
Repeated multiplication of a matrix and a vector
often occurs in applications.
For example, matrix A raised to the power of k
times vector x is equal to k factors of matrix A
times vector x.
This would be very difficult to do by hand,
or at least very tedious.
However, if lambda is an eigenvalue for matrix A
with matrix A times vector x equals lambda times vector x,
then matrix A raised to the power of k
times vector x is equal to lambda,
some scaler, raised to the power of k times vector x,
which obviously would be much easier to calculate
than the product above.
In the next lesson we'll take a look at how to determine
eigenvalues of a given matrix,
as well as how to determine the corresponding eigenvectors.
And I hope you found this helpful.