An eigenvector
of a matrix
is such that if the vector is multiplied by the matrix, the result is a multiple of the vector. Eigenvectors are special and have many applications in may areas. We ,ay write
In this equation
is a constant called the eigenvalue.
The procedure for finding eigenvectors and eigenvalues is quite simple. If
then
This means that the matrix
has zero determinant. We can solve
and solve this equation to find values of
In general several values of %lambda may be found. Each value ofgives value to at least one eigenvector, and different eigenvectors give rise to different eigenvalues.
Example: Find the eigenvalues and eigenvectors for the matrix
We obtain
Solving this gives
or
We find the eigenvectors
by solving
For
Hence
and an eigenvector is
For
Hence
and an eigenvector is
Notice that any vectors of the forms
and
are eigenvectors. We choose values of
to make the eigenvectors as simple as possible.