Section 4.3 Symmetric and Orthogonal Matrices and Quadratic Forms
Theorem 4.3.2.
The eigenvalues of a symmetric matrix are real.
Proof.
\begin{equation*}
A \vec{v} = \lambda \vec{v}
\end{equation*}
Right multiply by \(\vec{v}^T\)
\begin{equation*}
\vec{v}^T A \vec{v} = \lambda \vec{v}^T \vec{v}
\end{equation*}
Take the complex conjugate of the eigenvector equation above:
\begin{align*}
\overline{\vec{v}}^T A^T \amp = \overline{\lambda} \overline {\vec{v}}^T \\
\overline{\vec{v}}^T A \amp = \overline{\lambda} \overline {\vec{v}}^T
\end{align*}
left mulitply by \(\vec{v}\)
\begin{align*}
\overline{\vec{v}}^T A \vec{v} \amp = \overline{\lambda} \overline{\vec{v}}^T
\vec{v} \\
\overline{\vec{v}}^T \lambda \vec{v} \amp = \overline{\lambda}
\overline{\vec{v}}^T
\vec{v} \\
\lambda \overline{\vec{v}}^T \vec{v} \amp = \overline{\lambda}
\overline{\vec{v}}^T
\vec{v}
\end{align*}
so \(\lambda = \overline{\lambda}\text{,}\) hence it is real.
Theorem 4.3.3.
The eigenvectors of a symmetric matrix are orthogonal.
Definition 4.3.4.
A square matrix \(Q=[\vec{q}_1\;\; \vec{q}_2\;\;\vec{q}_3\;\; \cdots\;\;\vec{q}_n] \) is said to be orthogonal if
\begin{equation*}
\vec{q}_i^T \vec{q}_j = 0
\end{equation*}
for all \(i \neq j\text{.}\) If in addition that \(\vec{q}_i^T\vec{q}_i = 1\) for all \(i\) then the matrix is also said to be orthonormal.
Theorem 4.3.5.
Subsection 4.3.1 Orthogonal Transformations
Definition 4.3.6.
Let \(Q\) be a linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^n\text{.}\) If \(Q\) is an orthonormal matrix, then the linear transformation is said to be an orthogonal transformation.
Example 4.3.7.
Show that the rotational transformation given in section 7.10 is an orthogonal transformation.
Definition 4.3.8.
A quadratic form is a polynomial function in \(x_1, x_2, \ldots, x_n\) such that
\begin{equation*}
q(\vec{x}) = a_1 x_1^2 + a_2 x_2^2 + \cdots a_n x_n^2 + b_{12} x_1 x_2 + b_{13} x_1
x_3
+
\cdots
\end{equation*}
Note that a quadratic form can have squared terms and products between only two variables.
Example 4.3.9.
The following are examples of quadratic forms.
-
\(\displaystyle 4x_1^2 + 7x_2^2 + 4x_1x_2\)
-
\(\displaystyle x_1^2 + 4x_2^2 + 9 x_3^2 -2x_1x_2 + 3x_1x_3 + x_2x_3 \)
And the following are not:
-
\(x_1^2 + 4x_2^2 + 9x_2+ 3\text{.}\)
-
\(x_1^2 + 4x_2^2 + 3x_3^2 + 2x_1x_2 -3x_1 x_2 x_3 \text{.}\)
A quadratic form can be written as
\begin{equation*}
q(\vec{x}) = \vec{x}^T A \vec{x}
\end{equation*}
Example 4.3.10.
Find the matrix \(A\) for the quadratic forms above:
Solution.
\begin{align*}
A \amp= \begin{bmatrix}
4 \amp2 \\
2 \amp 7
\end{bmatrix}\\
\vec{x}^T A \vec{x} \amp = [x_1 \;\; x_2] \begin{bmatrix}
4 \amp2 \\
2 \amp 7
\end{bmatrix} \begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}\\
\amp = [x_1 \;\; x_2] \begin{bmatrix}
4x_1 + 2 x_2 \\
2 x_1 + 7 x_2
\end{bmatrix}\\
\amp = 4x_1^2 + 2 x_1 x_2 + 2 x_1 x_2 + 7 x_2^2
\end{align*}
And for the second quadratic form:
\begin{equation*}
A = \begin{bmatrix}
1 \amp -1 \amp 3/2 \\
-1 \amp 4 \amp 1/2 \\
3/2 \amp 1/2 \amp 9
\end{bmatrix}
\end{equation*}
Subsection 4.3.2 Principle Axes of Quadratic Forms
For the quadratic form:
\begin{equation*}
q(\vec{x}) = \vec{x}^T A \vec{x}
\end{equation*}
if \(A = P D P^{-1}\) can be written:
\begin{equation*}
q(\vec{x}) = \vec{x}^T P D P^{-1} \vec{x}
\end{equation*}
and since \(A\) is symmetric then the eigenvectors form an orthonormal basis so \(P^{-1} = P^T\text{.}\)
\begin{equation*}
q(\vec{x}) =\vec{y}^T D \vec{y}
\end{equation*}
where \(\vec{y}= P^T \vec{x}\text{.}\) Example of ellipse/contour plot.
