Let $\boldsymbol{V}\_{1},\dots,\boldsymbol{V}\_{n}\in\mathbb{R}^{d\times m}$ be $n$ “tall” matrices (where $d\ge m$) with orthonormal columns.


And let $\boldsymbol{P}\_{1},\dots,\boldsymbol{P}\_{n}\in\mathbb{R}^{d\times d}$ be the orthogonal projection matrices defined as $\boldsymbol{P}\_{i}=\boldsymbol{V}\_{i}\boldsymbol{V}\_{i}^{\top}$.


Finally, let $\boldsymbol{T}$ be an operator defined the concatenation of the projection matrices $\boldsymbol{T}=\boldsymbol{P}\_{n}\cdots\boldsymbol{P}\_{1}$.


**Question:** does there exists a sequence $\boldsymbol{V}\_{1},\dots,\boldsymbol{V}\_{n}$ such that the concatenation operator $\boldsymbol{T}$ is nilpotent with index $k\ge3$? That is,


$$\boldsymbol{T}^{k-1}=\left(\boldsymbol{P}\_{n}\cdots\boldsymbol{P}\_{1}\right)^{k-1}\neq\boldsymbol{0}\_{d\times d},\,\,\,\,\,\,\,\boldsymbol{T}^{k}=\left(\boldsymbol{P}\_{n}\cdots\boldsymbol{P}\_{1}\right)^{k}=\boldsymbol{0}\_{d\times d}$$




---


**Special case (example):** when $n=3$ and $d=2$, we can choose projections such that $\boldsymbol{T}$ is a nilpotent operator with an index of $k=2$.


Choose $\boldsymbol{v}\_{1}=\left[0,1\right]^{\top},
\boldsymbol{v}\_{2}=\frac{1}{\sqrt{2}}\left[1,1\right]^{\top},\boldsymbol{v}\_{3}=\left[1,0\right]^{\top}$.


Then, the projection matrices are
$$\boldsymbol{P}\_{1}=\boldsymbol{v}\_{1}\boldsymbol{v}\_{1}^{\top}=\left[\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right],\,\,\,\boldsymbol{P}\_{2}=\frac{1}{2}\left[\begin{array}{cc}
1 & 1\\
1 & 1
\end{array}\right],\,\,\,\boldsymbol{P}\_{3}=\left[\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right]$$
And the concatenation of these matrices is
$$\boldsymbol{T}=\boldsymbol{P}\_{3}\boldsymbol{P}\_{2}\boldsymbol{P}\_{1}=\frac{1}{2}\left[\begin{array}{cc}
0 & 1\\
0 & 0
\end{array}\right]$$
which is a nilpotent matrix with an index of $k=2$ (i.e., $\boldsymbol{T}^{2}=\boldsymbol{0}\_{d\times d}$).



Can the concatenation of projection operators be nilpotent with an index k>=3?
I think your example is easily generalisable for any index. For example, let
$$
Q\_1=P\_1\oplus(1), 
Q\_2=P\_2\oplus(1),
Q\_3=P\_3\oplus(1)=(1)\oplus P\_1,
Q\_4=(1)\oplus P\_2,
Q\_5=(1)\oplus P\_3.
 $$
Then $Q\_5Q\_4Q\_3Q\_2Q\_1$ is nilpotent of index 3, and I think that a similar construction with $2n-1$ matrices of size $n$ gives nilpotence of index $n-1$.


