Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

Question

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was an invariant plane if all 3 equations were the same when trying to find the eigenvectors, but does this only happen when there is a repeated eigenvalue, or does it happen also when there are 3 distinct eigenvalues?

Answer 1

If $Av=\lambda v$ and $Aw=\lambda w$, then for any linear combination $\alpha v+\beta w$ we have $$ A(\alpha v+\beta w)=\alpha Av+\beta Aw=\alpha\lambda v+\beta\lambda w=\lambda(\alpha v+\beta w). $$ In words, a linear combination of eigenvectors for the same eigenvalue is again an eigenvector for that eigenvalue.

That said, it could happen that no such linearly independent $v$ and $w$ exist: let $$ A=\begin{bmatrix} 2&1&0\\0&2&0\\0&0&3\end{bmatrix}. $$ Then, while $2$ is a repeated eigenvalue, its eigenspace is one-dimensional.

Answer 2

For any $n \times n$ (complex) matrix $A$, there are always invariant subspaces of $\mathbb C^n$ of all dimensions $\le n$. If $A$ is upper triangular, the $k$-dimensional subspace spanned by the first $k$ standard unit vectors is invariant. And every square matrix is similar to an upper triangular matrix.

Answer 3

Note: I realized eigenplan is the space formed by the eigenvectors for a repeated eigenvalue . So the title you ask a bout the existence of a eigenplan, but in the text you speak a bout invariant hyperplane . Then an invariant hyperplane is not necessarily a eigenplan, the converse is yes. The answer to the question given in the text, is already indicated, even more in the answer you get better. But namely when a eigenvalu is multiple of multiplicity m, the dimension of the associated eigenspace to this eigenvalue is bounded below by 1 and superiorly by m. So all cases are possible, for example, look at the type matrix of 3x3 , of Jordan form matrix. if my remark is false I apologize for my understanding of English, thanks