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The algebraic multiplicity of an eigenvalue $\lambda$ is the number of times $\lambda$ appears as a root of the characteristic polynomial.

The geometric multiplicity of an eigenvalue $\lambda$ is dimension of the eigenspace of the eigenvalue $\lambda$.

Let us consider the linear transformation $T:\Bbb R^3 \to \Bbb R^3$ for simplicity. Suppose the characteristic polynomial of $T$ has the eigenvalue $\lambda$ as a repeated root, $2$ times. For example, if the eigenspace of the eigenvalue $\lambda$ were a line (one-dimensional), we could visualize $T$ as the transformation squishing or stretching all vectors on that line by an amount $\lambda$. But what is the geometric significance of the algebraic multiplicity $2$, in this case? Is there any underlying geometric intuition?

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  • $\begingroup$ @Lucas That's not necessarily true. $\endgroup$
    – user400242
    Commented Mar 3, 2018 at 5:39
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    $\begingroup$ The algebraic multiplicity is the dimension of the generalized eigenspace, if you're familiar with that. $\endgroup$
    – Chris
    Commented Mar 3, 2018 at 6:04
  • $\begingroup$ The set of all eigenvectors with eigenvalue $\lambda$ is a vector -supspace $S$ on which the transformation is a dilation of magnitude $\lambda$ centered at the origin if $\lambda>0, $ or a dilation centered at the origin of magnitude $|\lambda|$ followed by (or preceded by) a reflection in the origin if $\lambda<0, $ or the transformattion maps $S$ to $\{0\}$ if $\lambda=0.$ Moreover $S$ is the largest such vector-subspace. $\endgroup$ Commented Mar 3, 2018 at 6:09
  • $\begingroup$ @ChrisRandall Could you please elaborate that in an answer? $\endgroup$
    – user400242
    Commented Mar 3, 2018 at 6:24
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    $\begingroup$ @Blue you might find my discussion here to be useful. $\endgroup$ Commented Mar 3, 2018 at 9:05

1 Answer 1

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As you mentioned, the geometric multiplicity of an eigenvalue $\lambda$ is the dimension of its eigenspace. The algebraic multiplicity is the dimension of what's called the generalized eigenspace.

If you have a linear transformation, represented by a matrix A, and an eigenvalue $\lambda$, a generalized eigenvector is a vector, $v$ such that for some integer n, $$(A-\lambda I)^nv =0$$

Then, the space formed by taking all such generalized eigenvectors is called the generalized eigenspace and its dimension is the algebraic multiplicity of $\lambda$.

There's a nice discussion of the intuition behind generalized eigenvectors here.

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