Invariant Points
Any transformation
represented by a matrix
whose entries are numbers is linear. The matrix will send a line to a line. Sometimes it may send points on a line to some other point on the same line, and sometimes it will send a point to itself. In the second case, the point is said to be invariant.
Every transformation represented by a matrix has at least one invariant point – the origin, since if is the matrix representing T,
where
indicates the zero vector with every entry equal to 0. Suppose that
is an invariant of
so that
The above equation means thatis an eigenvector ofwith eigenvalue 1. Not all matrices have such eigenvalues, so this is a condition of a transformation having invariant points other than vec 0 . If such a vectorexists, any scalar multiple ofwill also be invariant since
This means that the eigenvector corresponding to an eigenvalue of 1 will define a line every point of which is an invariant point.
Example:
Suppose a transformationis represented by
The eigenvalues are the solutions to
If
the eigenvectorsare the solutions to
Hence
and an eigenvector is
but this is not invariant, because the eigenvalue is 3 so
If
the eigenvectorsare the solutions to
Hence
and an eigenvector is
but this is invariant, because the eigenvalue is 1 so
Also, any scalar multiple ofis invariant so in particular
which defines the line
is invariant.
