Operator interpretation of matrices

If $A$ is an $n \times n$ matrix, we can think of it as an operator or transformation acting on a column vector $X$ to produce $AX$ which is another column vector. With this interpretation equation (21) asks for those vectors $X$ which are transformed by $A$ into constant multiples of themselves [or equivalently into vectors which have the same direction but possibly different magnitude].

If case $A$ is an orthogonal matrix, the transformation is a rotation and explains why the absolute value of all the eigenvalues in such case are equal to one, since an ordinary rotation of a vector would not change its magnitude.

The ideas of transformation are very convenient in giving interpretations to many properties of matrices.