Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ be a real matrix. Use lagrange multipliers to find a closed form of the operator norm of A.
I've defined $f(v) = \lVert Av \rVert ^2$ and $g(v) = \lVert v \rVert ^2$. We need to find the maximum of $f$ under the constraint $g=1$ (the maximum exists since the unit circle is compact and $f$ is continuous).
By using Lagrange multipliers, since the minimum exists, there exists $\lambda \in \mathbb R$ such that the matrix \begin{bmatrix}a^2+c^2-\lambda&ab+cd\\ab+cd&b^2+d^2-\lambda\end{bmatrix} is singular. Computing its determinant results in the quadratic equation: $$ (ad-bc)^2 -(a^2+b^2+c^2+d^2)\lambda +\lambda^2 = 0$$
I'm getting lost in simplifying the roots of this polynomial (using the quadratic formula) and I was wondering if there is a better way to do this.
Thanks in advance!