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
A Level Maths Notes: FP4 – Eigenvalues and Eigenvectors
An eigenvectorof
a matrixis
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.