The matrix A=\left[\begin{array}{cccc}
0 &0 &-5 &-1\cr
4 &-4 &-1 &3\cr
0 &0 &-4 &0\cr
0 &0 &4 &0
\end{array}\right]
has two distinct real eigenvalues \lambda_1 < \lambda_2 .
Find the eigenvalues and a basis for each eigenspace.

The smaller eigenvalue\lambda_1 is and
a basis for its associated eigenspace is

\Bigg\lbrace
\left[\Rule{0pt}{4.8em}{0pt}\right. \left]\Rule{0pt}{4.8em}{0pt}\right. ,
\left[\Rule{0pt}{4.8em}{0pt}\right. \left]\Rule{0pt}{4.8em}{0pt}\right.
\Bigg\rbrace.

The larger eigenvalue\lambda_2 is and
a basis for its associated eigenspace is

\Bigg\lbrace
\left[\Rule{0pt}{4.8em}{0pt}\right. \left]\Rule{0pt}{4.8em}{0pt}\right.
\Bigg\rbrace.

The smaller eigenvalue

The larger eigenvalue

You can earn partial credit on this problem.