I am supposed to find the geometric meaning of eigenvalues/eigenvectors of certain matrices such as reflections about x= y, rotaions, shears, etc. How would I go about this? The first question is a reflection about x = y. Would this just mean that the eigenvalues and vectors are switched? I am not really sure about how to go about this question.
Find the lines through the origin which are sent into themselves (not necessarily pointwise) by the operator $T$. Any nonzero vector on such a line is an eigenvector. Conversely, every eigenvector lies on a line which is sent to itself by $T$.
For example: for a reflection, there are two lines which are sent into themselves: the axis of reflection, which is mapped identically into itself by $T$ (eigenvalue $1$), and the axis perpendicular to it, which is mapped into itself by a reflection through the origin (eigenvalue $-1$).