I'm reading my textbook and I'm really confused about geometric multiplicity. I've read the definition and they have given an example but I'm still lost. I've tried looking it up on other websites. This helped me make sense of algebraic multiplicity but I don't understand geometric multiplicity at all. I was wondering if someone could explain this as simply as possible.
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$\begingroup$The geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace. For example, let $$ A= \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $$ Then $\operatorname{char}_A(\lambda)=(\lambda-1)^2$ so the only eigenvalue of $A$ is $\lambda_1=1$. Since the degree of the $(\lambda-1)$ term of the characteristic polynomial is $2$, we say that $\lambda_1$ has algebraic multiplicity two.
Now, we wish to find all of the eigenvectors of $A$ corresponding to $\lambda_1$. That is, we want all vectors $v$ such that $Av=v$ (see if you can do this yourself). By inspecting the equation $Av=v$, we see that all of the eigenvectors of $A$ corresponding to $\lambda_1$ are of the form $$v=a\cdot\begin{pmatrix}1\\ 0\end{pmatrix}\qquad a\in\mathbb R$$ That is, the eigenspace of $A$ corresponding to $\lambda_1$ is $$ E_{\lambda_1}=\operatorname{Span}\left\{\begin{pmatrix}1\\ 0\end{pmatrix}\right\} $$ Hence $\dim E_{\lambda_1}=1$ and the geometric multiplicity of $\lambda_1$ is one (note that this is different from the algebraic multiplicity).
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