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

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

diagonalizationeigenvalues-eigenvectorslinear algebramatrices

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

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diagonalizationeigenvalues-eigenvectorslinear algebramatrices

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

## Best Answer

If $p$ is a polynomial and $\lambda \in \mathbb{C}$, then

algebraic multiplicityof $\lambda$ 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 polynomialof an $n$-by-$n$ matrix over an algebraically closed field (like $\mathbb{C}$) is always $n$.