MAT3040 Advanced Linear Algebra ·

Statement

A matrix is nilpotent if some positive integer power of it is zero.

The set WW of all nilpotent matrices in M2(R)M_2(\mathbb R), the space of real 2×22\times2 matrices, is a vector subspace of M2(R)M_2(\mathbb R).

Proposed proof

Let W={A∈M2(R):Ar=0 for some r≥1}. W=\{A\in M_2(\mathbb R):A^r=0\text{ for some }r\geq1\}. Clearly 0∈W0\in W. Let A,B∈WA,B\in W and α,β∈R\alpha,\beta\in\mathbb R. Choose positive integers r,sr,s such that Ar=0A^r=0 and Bs=0B^s=0. By the binomial theorem, (αA+βB)r+s−1=∑k=0r+s−1(r+s−1k)αkβr+s−1−kAkBr+s−1−k. \begin{aligned} & (\alpha A+\beta B)^{r+s-1}\\ &\quad=\sum_{k=0}^{r+s-1}\binom{r+s-1}{k} \alpha^k\beta^{r+s-1-k}A^kB^{r+s-1-k}. \end{aligned}

For each kk, either k≥rk\geq r, in which case Ak=0A^k=0, or k≤r−1k\leq r-1, in which case r+s−1−k≥s r+s-1-k\geq s and Br+s−1−k=0B^{r+s-1-k}=0. Thus every summand vanishes, giving (αA+βB)r+s−1=0. (\alpha A+\beta B)^{r+s-1}=0.

Hence αA+βB∈W\alpha A+\beta B\in W, so WW is a vector subspace. □\square