Resposta:

[tex]\lambda_{1} = -2\\\lambda_{2} = 6\\[/tex]

[tex]\bold{v}_{\lambda1} = (y,y)[/tex]

[tex]\bold{v}_{\lambda2} = (-y,y)[/tex]

Explicação passo a passo:

[tex]det\left[\begin{array}{ccc}2-\lambda&-4\\-4&2-\lambda\end{array}\right] =0[/tex]

Autovalores:

[tex](2-\lambda)^2-16=0\\\\\lambda_{1} = -2\\\lambda_{2} = 6\\[/tex]

Autovetores:

Para [tex]\lambda_1[/tex] = -2;  v = [tex]\left[\begin{array}{ccc}x\\y\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}2-(-2)&-4\\-4&2-(-2)\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

[tex]\left \{ {{4x-4y=0} \atop {-4x+4y=0}} \right. \\\\x=y[/tex]

Então, [tex]\bold{v}_{\lambda1} = (y,y)[/tex] sendo um de seus representantes o vetor [tex]\bold{v}_1 = (1,1)[/tex]

Para [tex]\lambda_2[/tex] = 6;  v = [tex]\left[\begin{array}{ccc}x\\y\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}2-6&-4\\-4&2-6\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

[tex]\left \{ {{-4x-4y=0} \atop {-4x-4y=0}} \right. \\\\x=-y[/tex]

Então, [tex]\bold{v}_{\lambda2} = (-y,y)[/tex] sendo um de seus representantes o vetor [tex]\bold{v}_2 = (-1,1)[/tex]