## 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 that is an eigenvector of with 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 vector exists, any scalar multiple of will 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 transformation is represented by The eigenvalues are the solutions to   If the eigenvectors are the solutions to  Hence and an eigenvector is but this is not invariant, because the eigenvalue is 3 so If the eigenvectors are the solutions to  Hence and an eigenvector is but this is invariant, because the eigenvalue is 1 so Also, any scalar multiple of is invariant so in particular which defines the line is invariant. 