I have to find a 3×3 matrix M with only one given eigenvalue λ = 0 and an eigenvector v = (1,1,1). These should be the only solutions for the matrix.
How do I do that? Which relations should I use?
Find 3×3 Matrix with specific eigenvalue and eigenvector
linear algebramatricesmatrix decomposition
Best Answer
If $(\lambda, x)$ is an eigenpair of $M$ then we indeed have $Mx = \lambda x.$ But $\lambda = 0$ in the given eigenpair, so we must have $Mx = 0.$ That is, $x \in \mathrm{Nul} \ M$ (the null space). Now we know the given eigenvector $v = (1, 1, 1)$ must be in this null space.
We should note that there are infinitely many $M$ that satisfy your given conditions. To get an equivalence class, think of the simple case where the matrix you want is in row echelon form. Form a simple relation with the entries of this eigenvector, say $2v_1 - v_2 - v_3 = 0$ and you'll have that a row in your matrix should be $(2, -1, -1).$ Thus, one such $M$ is
$$M = \begin{pmatrix} 2 & -1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}.$$
Check by multiplying and finding the eigenvalues!