It's another way of saying that T - λI is not a one-to-one map, meaning that (T - λI) is non-invertible. So there are vectors x and y such that (T - λI)x = (T -

It's another way of saying that T - λI is not a one-to-one map, meaning that (T - λI) is non-invertible. So there are vectors x and y such that (T - λI)x = (T - λI)y for y ≠ x. Which means that (T - λI)(x - y) = 0, and in general because of linearity every scalar multiple of (x - y) also maps to zero. Letting (x - y) = v, we could get (T - λI)v = 0 ==> Tv = λv, which is perhaps a more familiar definition. So, it's a neat way of expressing the concept, but I'm not sure what it buys you in terms of improving one's intuition.