= Solution
For $\mathbf u=(1,1)^T$ and $\mathbf v=(1,-1)^T$, the <outer products> are
$$
\mathbf u\mathbf u^T=\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad \mathbf v\mathbf v^T=\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
$$
Their <inner product> is zero and both squared <norms> are two. Thus $P_u=\mathbf u\mathbf u^T/2$ and $P_v=\mathbf v\mathbf v^T/2$ are orthogonal <spectral projectors>, with $P_u+P_v=I$. Since $A\mathbf u=5\mathbf u$ and $A\mathbf v=\mathbf v$, the <eigenvalues> are $\lambda=5$, $\mu=1$, and
$$
\boxed{A=\frac52\mathbf u\mathbf u^T+\frac12\mathbf v\mathbf v^T,\qquad A^{-1}=\frac1{10}\mathbf u\mathbf u^T+\frac12\mathbf v\mathbf v^T.}
$$
Multiplying these decompositions uses $P_uP_v=0$ and $P_u^2=P_u$, $P_v^2=P_v$, giving the <identity matrix>.
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