There are several similar questions:
(a) If possible, write down a 5 × 5 real matrix with −1 as its only real eigenvalue and where the eigenspace with eigenvalue −1 has dimension 3.
(b) If possible, write down a 5 × 5 real matrix with −1 as its only real eigenvalue and where the eigenspace with eigenvalue −1 has dimension 5.
(c) If possible, write down a 5 × 5 real matrix with −1 as its only real eigenvalue and where the eigenspace with eigenvalue −1 has dimension 1 and the generalized eigenspace with eigenvalue −1 has dimension 3.
(d) If possible, write down a 5 × 5 real matrix with −1 as its only real eigenvalue and where the eigenspace with eigenvalue −1 has dimension 1 and the generalized eigenspace with eigenvalue −1 has dimension 4.
Can anyone give me some hints for each one or provide a general way to slove all of them and other similar questions?
I will answer the first question. Try to adapt the answer to the other questions.
An example of a matrix that satisfies the property mentioned in (a) is$$\begin{bmatrix}-1&0&0&0&0\\0&-1&0&0&0\\0&0&-1&0&0\\0&0&0&0&-1\\0&0&0&1&0\end{bmatrix}.$$