Linear Algebra

Section III. Nilpotence 3792.9 Example The differentiation map d/dx: P 2 → P 2 is nilpotent of index threesince the third derivative of any quadratic polynomial is zero. This map’s actionis described by the string x 2 ↦→ 2x ↦→ 2 ↦→ 0 and taking the basis B = 〈x 2 , 2x, 2〉gives this representation.⎛⎜0 0 0⎞⎟Rep B,B (d/dx) = ⎝1 0 0⎠0 1 0Not all nilpotent matrices are all zeros except for blocks of subdiagonal ones.2.10 Example With the matrix ˆN from Example 2.5, and this four-vector basis⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 0 1 00D = 〈 ⎜ ⎟⎝1⎠ , 2⎜ ⎟⎝1⎠ , 1⎜ ⎟⎝1⎠ , 0⎜ ⎟⎝0⎠ 〉0 0 0 1a change of basis operation produces this representation with respect to D, D.⎛⎞ ⎛⎞ ⎛ ⎞1 0 1 0 0 0 0 0 1 0 1 00 2 1 01 0 0 00 2 1 0⎜⎟ ⎜⎟ ⎜ ⎟⎝1 1 1 0⎠⎝0 1 0 0⎠⎝1 1 1 0⎠0 0 0 1 0 0 1 0 0 0 0 1−1⎛⎞−1 0 1 0−3 −2 5 0= ⎜⎟⎝−2 −1 3 0⎠2 1 −2 0The new matrix is nilpotent; it’s fourth power is the zero matrix. We couldverify this with a tedious computation, or we can observe that it is nilpotentsince it is similar to the nilpotent matrix ˆN 4 .(P ˆNP −1 ) 4 = P ˆNP −1 · P ˆNP −1 · P ˆNP −1 · P ˆNP −1 = P ˆN 4 P −1The goal of this subsection is to show that the prior example is prototypicalin that every nilpotent matrix is similar to one that is all zeros except for blocksof subdiagonal ones.2.11 Definition Let t be a nilpotent transformation on V. A t-string of lengthk generated by ⃗v ∈ V is a sequence 〈⃗v, t(⃗v), . . . , t k−1 (⃗v)〉. A t-string basis is abasis that is a concatenation of t-strings.Note that the strings cannot form a basis under concatenation if they arenot disjoint because a basis cannot have a repeated vector.2.12 Example In Example 2.6, we can concatenate the t-strings 〈⃗β 1 , ⃗β 2 , ⃗β 3 〉 and〈⃗β 4 , ⃗β 5 〉, of length three and two, to make a basis for the domain of t.2.13 Lemma If a space has a basis of t-strings then the longest string in thatbasis has length equal to the index of nilpotency of t.