Linear Algebra

Section III. Nilpotence 385and finish by adding ⃗β 1 ∈ N (n 3 ) = C 5 such that n(⃗β 1 ) = ⃗β 2 .Exercises⎛ ⎞10⃗β 1 =1⎜ ⎟⎝0⎠0̌ 2.18 What is the index of nilpotency of the left-shift operator, here acting on thespace of triples of reals?(x, y, z) ↦→ (0, x, y)̌ 2.19 For each string basis state the index of nilpotency and give the dimension ofthe range space and null space of each iteration of the nilpotent map.(a) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗0⃗β 3 ↦→ ⃗β 4 ↦→ ⃗0(b) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗β 3 ↦→ ⃗0⃗β 4 ↦→ ⃗0⃗β 5 ↦→ ⃗0⃗β 6 ↦→ ⃗0(c) ⃗β 1 ↦→ ⃗β 2 ↦→ ⃗β 3 ↦→ ⃗0Also give the canonical form of the matrix.2.20 Decide which of these matrices are nilpotent.⎛ ⎞( ) ( ) −3 2 1−2 4 3 1(a)(b)(c) ⎝−3 2 1⎠−1 2 1 3−3 2 1⎛⎞45 −22 −19(e) ⎝33 −16 −14⎠69 −34 −29̌ 2.21 Find the canonical form of this matrix.⎛⎞0 1 1 0 10 0 1 1 10 0 0 0 0⎜⎟⎝0 0 0 0 0⎠0 0 0 0 0(d)⎛1 1⎞4⎝3 0 −1⎠5 2 7̌ 2.22 Consider the matrix from Example 2.17.(a) Use the action of the map on the string basis to give the canonical form.(b) Find the change of basis matrices that bring the matrix to canonical form.(c) Use the answer in the prior item to check the answer in the first item.̌ 2.23 Each of these matrices is nilpotent.⎛ ⎞( ) 0 0 01/2 −1/2(a)(b) ⎝0 −1 1⎠1/2 −1/20 −1 1Put each in canonical form.(c)⎛−1 1⎞−1⎝ 1 0 1⎠1 −1 12.24 Describe the effect of left or right multiplication by a matrix that is in thecanonical form for nilpotent matrices.2.25 Is nilpotence invariant under similarity? That is, must a matrix similar to anilpotent matrix also be nilpotent? If so, with the same index?