Invariant Points

Any transformationrepresented by a matrixwhose 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,whereindicates the zero vector with every entry equal to 0. Suppose thatis an invariant ofso 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 byThe eigenvalues are the solutions to

Ifthe eigenvectorsare the solutions to

Henceand an eigenvector isbut this is not invariant, because the eigenvalue is 3 so

Ifthe eigenvectorsare the solutions to

Henceand an eigenvector isbut this is invariant, because the eigenvalue is 1 so

Also, any scalar multiple ofis invariant so in particularwhich defines the lineis invariant.