Recall from the hint that a matrix
will have first column
, and second column
, where
and
are the standard basis vectors
and
.
Now, note that
, and use the property that
for an eigenvalue-eigenvector pair to get
![{\displaystyle A{\vec {e}}_{1}=A{\vec {v}}_{1}=\lambda _{1}{\vec {v}}_{1}=\lambda _{1}{\vec {e}}_{1}={\vec {e}}_{1},}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/5a3a6d47b036e8a3f2b0c321da39435cac8d932d)
since
. This results in the first column of
simply being
.
For the second column of
, we first note that
. Using the same eigenvalue-eigenvector relation we have
![{\displaystyle A{\vec {e}}_{2}=A({\vec {v}}_{2}-2{\vec {v}}_{1})=A{\vec {v}}_{2}-2A{\vec {v}}1=\lambda _{2}{\vec {v}}_{2}-2\lambda _{1}{\vec {v}}_{1}=2{\vec {v}}_{2}-2{\vec {v}}_{1},}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/02f96de1cea2e11bfa3eb8d98e7c182da7a6b699)
since
. This results in the second column being
, and the matrix is
![{\displaystyle A=\left[{\begin{array}{cc}1&2\\0&2\end{array}}\right].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/c455ed70c2264117f247700b25e7a98fb307e752)