# When is the geometric multiplicity of an eigenvalue smaller than its algebraic multiplicity?

I was kinda crushed to discover that two different matrices with different properties can actually share the same characteristic polynomial ($-\lambda^3-3\lambda^2+4$):

$A=\begin{pmatrix} 1 & 2& 2\\ -3 &-5 &-3 \\ 3& 3 & 1 \end{pmatrix} , B=\begin{pmatrix} 2 & 4& 3\\ -4 &-6 &-3 \\ 3& 3 & 1 \end{pmatrix}$

$A$ has an eigenline and an eigenplane (and thus an eigenbasis), whereas $B$ has two eigenlines (so no eigenbasis). The repeated eigenvalue -2 of B corresponds to an eigenspace with basis {(-1,1,0)}.

When is the geometric multiplicity of an eigenvalue smaller than its algebraic multiplicity (as in case B)? Are there general conditions to look for?

Thanks!

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When you matrix is not diagonalisable –  M Turgeon Aug 6 '12 at 19:16
@M I know this result/theorem, but I'm hoping for a more fundamental answer. Like some algebraic property or somesuch that will maybe unfurl during row operations. Or something. –  Ryan Aug 6 '12 at 19:22
It's simple to build examples of matrices with repeated eigenvalues whose behavior with respect to diagonalization differs, as Qiaochu alludes to. To use $3\times 3$ matrices as an example, $$\begin{pmatrix}\lambda_1&1&\\&\lambda_1&\\&&\lambda_2\end{pmatrix}$$ will only have two eigenvectors (i.e. it is a defective matrix), while $$\begin{pmatrix}\lambda_1&&\\&\lambda_1&\\&&\lambda_2\end{pmatrix}$$ is certainly diagonalizable. –  Ｊ. Ｍ. Aug 7 '12 at 0:23
@JM Ah, thanks for the concrete example and link:) –  Ryan Aug 7 '12 at 4:05

The general condition is the presence of nontrivial Jordan blocks.

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