With the injectivity-based definition, you can ask "how injective is it?" in various ways (eg, what is the dimension of the nullspace of T - λI; the Ax = λx def

With the injectivity-based definition, you can ask "how injective is it?" in various ways (eg, what is the dimension of the nullspace of T - λI; the Ax = λx definition is rather binary and corresponds to simply "T - λI has nullity > 0"). This is what is meant by "eigenvalue multiplicity" (there are actually at least two different inequivalent ways to measure this, geometric multiplicity and algebraic multiplicity). Also for finite-dimensional vector space a lot of concepts are equivalent, eg "a linear operator L is invertible <=> L is injective", which are not in infinite-dimensional vector spaces (eg https://math.stackexchange.com/a/2447563/21437 https://math.stackexchange.com/a/2447563/21437). Studying T - λI turns out to be more useful in this case (eg https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis) https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(fun...)