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Projections and Nilpotents: Maps with Memory

Two families of operators reveal structure through what happens when you apply them twice. Projections satisfy P^2 = P and split the space in two; nilpotents satisfy N^k = 0 and shred it. Both are coordinate-free and both build intuition for everything later.

Projections: doing it twice changes nothing

A projection is an operator P: V -> V with P^2 = P. Once you have projected, projecting again does nothing — the output is already settled. An operator obeying X^2 = X is called idempotent, and idempotent is exactly the algebraic fingerprint of a projection. This generalizes the orthogonal projection from Vol I, but here P need not be orthogonal.

The payoff: every projection splits V into a direct sum V = im P ⊕ ker P. The image is exactly the vectors P leaves fixed (P fixes everything it lands on, since P(Pv) = Pv), and the kernel is what P erases. Every v decomposes uniquely as v = Pv + (v - Pv), the first piece in the image, the second in the kernel.

P on R^2,  P = [1, 1;
               0, 0]

Check idempotent: P^2 = [1,1; 0,0] * [1,1; 0,0]
                      = [1,1; 0,0] = P   OK

Image  = { (a, 0) }      (the x-axis)   -> P fixes these
Kernel = { (t, -t) }     (the line y=-x) -> P kills these

Split:  (3, 2) = P(3,2) + (3,2 - P(3,2))
              = (5, 0) + (-2, 2)
         image piece + kernel piece, sum back to (3,2)
A skew (non-orthogonal) projection onto the x-axis along the line y = -x.

Nilpotents: eventually everything dies

At the opposite extreme sits the nilpotent operator: N with N^k = 0 for some power k. Apply it enough times and it annihilates every vector. The cleanest model is the shift that pushes a basis down a chain and drops the last rung off the end.

N on R^3 (the shift):  N(e1)=0,  N(e2)=e1,  N(e3)=e2

Matrix:  N = [0, 1, 0;
              0, 0, 1;
              0, 0, 0]

Iterate:
  N^1 sends e3 -> e2 -> e1 -> 0
  N^2 = [0,0,1; 0,0,0; 0,0,0]   (still nonzero)
  N^3 = 0                       (everything dies)

So k = 3 is the nilpotency index. The kernels grow:
  ker N   = span(e1)
  ker N^2 = span(e1, e2)
  ker N^3 = all of R^3
The basic nilpotent: a single shift chain whose kernels climb one rung at a time.

Nilpotents are the raw material of the Jordan form you will meet later: every operator, after subtracting its eigenvalue, looks nilpotent on each generalized eigenspace. Notice too that N restricted to ker N^2 is still nilpotent — nilpotency survives restriction to an invariant subspace, a fact that drives the whole structure theory.

Why these two are templates

Projections and nilpotents are at opposite poles of operator behavior. A projection is as stable as possible — it repeats itself forever. A nilpotent is as transient as possible — it self-destructs. Most operators are built from blends of these two flavors, which is exactly why mastering them now pays off when we decompose general operators into invariant pieces.