1. Symmetric matrices

As you know, even for a $2 \times 2$ matrix $A$, the properties of $A$ might not be ideal. There might not be a basis of linearly independent eigenvectors, or the eigenvalues and eigenvectors might be complex, or perhaps the eigenspaces of $A$ are not perpendicular. In contrast, symmetric matrices are guaranteed to have the best possible properties:

Theorem (``spectral theorem''). If $A$ is a real symmetric matrix, then

(1) the eigenvalues of $A$ are real numbers,

(2) the eigenspaces of $A$ are perpendicular to one another,

(3) $A$ can be diagonalized using a rotation matrix; in other words, $R ^ {-1} A R = D$ for some rotation matrix $R$.

The name ``spectral theorem'' comes from advanced applications in physics where the eigenvalues determine frequencies, like a color spectrum of light.

Problem DD-1. Explain how (3) implies (1) and (2). (Method: In (3), does $D$ have real entries? Are the eigenvectors perpendicular?)

The nice thing in applications is that there is is not much to do to find $R$; as the theorem says, the eigenspaces are automatically perpendicular to one another.

Example: $A = \left[\begin{array}{cc}3&1\\ 1&3\end{array}\right]$. Here in the past we found that we could use $P = \left[\begin{array}{rr}1&1\\ 1&-1\end{array}\right]$ and $D = \left[\begin{array}{rr}4&0\\ 0&2\end{array}\right]$. But $P$ is not a rotation matrix, since it violates both (i) and (ii). The columns are perpendicular as expected, though. To fix (i), find the length of each column and divide by it to make all column lengths be 1:

$\left[\begin{array}{rr}\frac 1{\sqrt 2}& \frac 1{\sqrt 2}\\ \frac 1{\sqrt 2}& -\frac 1{\sqrt 2}\end{array}\right]$.

Now check the determinant. It's $-1$, so change the sign of a column, say the second. We get

$R = \left[\begin{array}{rr}\frac 1{\sqrt 2}& -\frac 1{\sqrt 2}\\ \frac 1{\sqrt 2}& \frac 1{\sqrt 2}\end{array}\right]$.

Now $R$ does fit the definition of a rotation. Notice that $R = R _ {45 ^ \circ}$.

Problem DD-2. Diagonalize $A = \left[\begin{array}{rr}6&2\\ 2&9\end{array}\right]$ with a rotation matrix.