The characteristic equation for a second-order matrix \(M\) is a quadratic
equation. We have considered in detail the case in which M is a real symmetric
matrix and the roots of the characteristic equation (eigenvalues) are real,
positive, and unequal. Discuss some other possibilities as follows:
(a) \(\quad \mathrm{M}\) real and symmetric, eigenvalues real, one positive and
one negative. Show that the plane is reflected in one of the eigenvector lines
(as well as stretched or shrunk). Consider as a simple special case
$$M=\left(\begin{array}{rr}
1 & 0 \\
0 & -1
\end{array}\right)$$
(b) \(\quad \mathrm{M}\) real and symmetric, eigenvalues equal (and therefore
real). Show that \(\mathrm{M}\) must be a multiple of the unit matrix. Thus show
that the deformation consists of dilation or shrinkage in the radial direction
(the same in all directions) with no rotation (and reflection in the origin if
the root is negative).
(c) \(\quad M\) real, not symmetric, eigenvalues real and not equal. Show that
in this case the eigenvectors are not orthogonal. Hint: Find their dot
product.
(d) \(\quad \mathrm{M}\) real, not symmetric, eigenvalues complex. Show that all
vectors are rotated, that is, there are no (real) eigenvectors which are
unchanged in direction by the transformation. Consider the characteristic
equation of a rotation matrix as a special case.
