Exercise 2

For each string basis state the index of nilpotency and give the dimension of the range space and null space of each iteration of the nilpotent map.

• $\begin{array}{ccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{0}\\ \vec{\beta}_{3}&\mapsto&\vec{\beta}_{4}&\mapsto&\vec{0}\end{array}$

• $\begin{array}{ccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{\beta}_{3}&\mapsto&\vec{0}\\ \vec{\beta}_{4}&\mapsto&\vec{0}\\ \vec{\beta}_{5}&\mapsto&\vec{0}\\ \vec{\beta}_{6}&\mapsto&\vec{0}\end{array}$

• $\begin{array}{ccccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{\beta}_{3}&\mapsto&\vec{0}\end{array}$

Also give the canonical form of the matrix.