The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue.
Here are my questions:
- Where is the name "geometric multiplicity" from in math history?
- Is there any thing in geometry related to this concept? Does it only mean the number of linearly independent eigenvectors with that eigenvalue one can "draw" as the definition says?
Best Answer
@Bruno is essentially correct. It's important to see that geometric multiplicity is meant to be distinguished from algebraic multiplicity of eigenvalues, the latter being the total number of times an eigenvalue occurs as a root of the characteristic equation. An analogy can be made with roots of any polynomial. For example, $x^2 + 2x + 1$ has a single root $-1$ of multiplicity 2. Algebraically, there are always two roots for a quadratic (at least over $\mathbb{C}$), and in this case, those roots are $-1$ and $-1$. But (geometrically) there is only one $x$-intercept for the function $y = x^2 + 2x + 1$.