Can $n\times n$ matrix have algebraic multiplicity less than $n$?

Question

I can't seem to think of an example that would have an algebraic multiplicity less than $n$.

Answer 1

If $p$ is a polynomial and $\lambda \in \mathbb{C}$, then algebraic multiplicity of $\lambda$ is the maximum number of times the factor $(t-\lambda)$ appears when $p$ is decomposed into linear factors.

Thus, applying 'algebraic multiplicity' to matrices is not appropriate.

The degree of the characteristic polynomial of an $n$-by-$n$ matrix over an algebraically closed field (like $\mathbb{C}$) is always $n$.

Answer 2

Assuming that you mean the total number of eigenvalues of a given matrix over a given Field as the algebraic multiplicity of a matrix, consider $$ \begin {pmatrix} 0&-1\\ 1&0\\ \end {pmatrix} $$ This matrix has no eigenvalues in the real Field. As already pointed out, the concept of algebraic multiplicity for a matrix does not exist.