Real matrix/2x2/Trigonalizability via characteristic polynomial/Example

We consider a real ${\displaystyle {}2\times 2}$-matrix ${\displaystyle {}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$. The characteristic polynomial is

${\displaystyle {}{\begin{aligned}\chi _{M}&=\det {\left(xE_{2}-M\right)}\\&=\det {\begin{pmatrix}x-a&-b\\-c&x-d\end{pmatrix}}\\&=(x-a)(x-d)-bc\\&=x^{2}-(a+d)x+ad-bc\\&={\left(x-{\frac {a+d}{2}}\right)}^{2}-{\left({\frac {a+d}{2}}\right)}^{2}+ad-bc\\&={\left(x-{\frac {a+d}{2}}\right)}^{2}-{\left({\frac {a-d}{2}}\right)}^{2}-bc.\end{aligned}}}$

This polynomial splits into (real) linear factors if and only if ${\displaystyle {}{\left({\frac {a-d}{2}}\right)}^{2}+bc\geq 0}$. The matrix is trigonalizable exactly in this case, due to fact.