Suppose that $T:\mathbb{R}^2 \to \mathbb{R}^2$ has representing matrix $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathfrak{M}_2(\mathbb{R})$$ with respect to the standard basis in $\mathbb{R}^2$. Put $$\alpha:=\frac{1}{2}(a^2+b^2+c^2+d^2)$$ $$\beta:=\frac{1}{2}(a^2-b^2+c^2-d^2)$$ $$\gamma:=ab+cd$$ Show that $$\|T\|=\sqrt{\alpha^2+\sqrt{\beta^2+\gamma^2}}.$$
Given a point $x=(x_1,x_2)\in\mathbb{R}^2$, then $$T(x_1,x_2)=(ax_1+bx_2,cx_1+dx_2)$$ Using the usual norm in $\mathbb{R}^2$, we have that $$\|T(x)\|_2=\sqrt{(ax_1+bx_2)^2+(cx_1+dx_2)^2}$$ Using that, for the norm of the operator, we evaluate just points with $\|x\|=1=x_1^2+x_2^2$, then is possible to rewrite the norm of the operator in each point as $$\|T(x)\|_2=\sqrt{(a^2-b^2+c^2-d^2)x_1^2+2(ab+cd)x_1\sqrt{1-x_1^2}}$$
How should I proceed from this point?