1. λI stretches every vector (the whole space, really) by a factor λ.
2. Saying that the function is not injective means you lose information: when you apply it on some object and get a result, you can't trace back what was the original object, as there may be several. (There is no inverse function, then). In linear algebra, this only happens because there is some direction of space where all the vectors get collapsed to zero.
In short, T-λI collapses some line of vectors to zero.
So, when you took the effect of λI from T, you make it a lossy transformation in some direction. This means that _in that direction_ T had the effect of stretching all vectors by a factor of λ.
You gain some geometric understanding of T.
It is sort of intuitive, but the language may obscure it a little if you are not used to it.
If I understand right, you’re saying that there’s an interpretation in terms of the geometry of the T transformation, of subtracting this diagonal matrix from T. Multiplication of matrices is composition of transformations, I get that, but I’m not so sure what adddition/subtraction is.
Yes, that's right. Addition is just applying the transformations separately to the same vector and adding the result. So what this is saying is that if you apply λI to a vector in that particular direction, then there is nothing left to add to get the effect of T.
Ideally you would like to do this for all n directions of space, and that way you completely describe what T does in simpler terms: it just stretches things differently in different directions. It's not always possible though. The matrices that allow this are called diagonalizable and the process of finding the stretch factors (eigenvalues) is called diagonalization.
Just a caveat: if an eigenvalue is complex, the effect is not as simple as a stretch, but the interpretation is very similar.
1. λI stretches every vector (the whole space, really) by a factor λ.
2. Saying that the function is not injective means you lose information: when you apply it on some object and get a result, you can't trace back what was the original object, as there may be several. (There is no inverse function, then). In linear algebra, this only happens because there is some direction of space where all the vectors get collapsed to zero.
In short, T-λI collapses some line of vectors to zero.
So, when you took the effect of λI from T, you make it a lossy transformation in some direction. This means that _in that direction_ T had the effect of stretching all vectors by a factor of λ.
You gain some geometric understanding of T.
It is sort of intuitive, but the language may obscure it a little if you are not used to it.