How can one solve this optimization problem by rewriting the matrix?

Question

Given this problem:

$$maximize \, \, x^T\begin{bmatrix} 3 & 4 \\ 0 & 3 \end{bmatrix}x $$

$$ subject \,to \, \, ||x||^2 = 1$$

I want to solve this using Langrange Conditions. What I wish to do is find the eigenvalues of a symmetric matrix $Q$, and the corresponding eigenvectors to maximize this problem.

In the solution material the textbook author rewrites the matrix: $\begin{bmatrix} 3 & 4 \\ 0 & 3 \end{bmatrix} $ into $ Q = \begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$ , where $Q^T = Q$. How and why can one do this?

(Afterwards one can calculate the eigenvalues of $Q$ and find the corresponding eigenvectors)

Thank you.

Answer 1

That manipulation is valid since $$x^T\begin{bmatrix}3&4\\0&3\end{bmatrix}x = 3x_1^2+4x_1x_2+3x_2^2 = 3x_1^2+2x_1x_2+2x_2x_1+3x_2^2 = x^T\begin{bmatrix}3&2\\2&3\end{bmatrix}x.$$

In general, if $A \in \mathbb{R}^{n \times n}$ and $x \in \mathbb{R}^n$, then $x^TAx$ is a scalar. Hence, $x^TAx = (x^TAx)^T = x^TA^Tx$, and thus, $$x^TAx = \dfrac{1}{2}x^TAx+\dfrac{1}{2}x^TA^Tx = x^T\left(\dfrac{A+A^T}{2}\right)x.$$