Find matrix with given eigenvalues

Question

I am given all eigenvalues of a matrix: $1$,$2$, and $4$ and I have to find the matrix that has these eigenvalues.

What I concluded so far is that every upper-triangular and lower-triangular matrix with these values at the diagonal can be the matrix I'm looking for, but I struggle to find a general rule that will give me every matrix with these eigenvalues.

Answer 1

This assumes you only want 3 by 3 matrices. If not the answer is a little bit more messy.

Let $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$.

Then the collection of matrices that satisfy your condition is $ADA^{-1}$ where $A$ is any invertible 3 by 3 matrix.

Answer 2

Note that similar matrices have the same eigenvalues.

Start with a diagonal $3$ by $3$ matrix,$D$ with your eigenvalues on the main diagonal.

Choose an arbitrary invertible matrix $P$ and construct $ M=PDP^{-1} $

This M is your solution.

Answer 3

Every matrix with those eigenvalues will have a Jordan Canonical Form with those eigenvalues. So you just have to take every JDF with those eigenvalues, and conjugate them.