I was studying generalized eigenvalues, and I read the following property of the algebraic multiplicity of an eigenvalue $\lambda$: the algebraic multiplicity of $\lambda$ is the maximum number of linearly independent generalized $\lambda$-eigenvectors.
My question is on the use of the word "maximum". Is this saying that that the number of linearly independent generalized $\lambda$-eigenvectors can be at most the algebraic multiplicity, or is it saying that it has exactly this number?
It means that it is possible to construct a set of linearly independent eigenvectors with size equal to the algebraic multiplicity of the eigenvalue, but not more than it. You can always construct a set with fewer linearly independent eigenvectors, because all you do is take that maximal set and remove some of them.