Linear manifolds of projectors

A. M. Vetoshkin

Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 42-55

Résumé

The paper shows that a linear manifold of matrices of the form: $Q=Q_{0}+\sum a_{i}P{}_{i}$, can consist of projectors only. It turns out that for this it is necessary and sufficient that $P_{i} =Q_{i}-Q_{0}$ and all the matrices $Q_{i}$ be projectors, moreover: $(Q_{i}-Q_{j})^{2}=0$ for any pair i and j. It is established that all projectors that make up this linear manifold have one rank and any pair $A, B$ of these projectors satisfies $(A-B)^{2}=0$. Several conditions were found equivalent to the fact that two projectors $A,B$ satisfy $(A-B)^{2}=0$, one of them in terms of the subspaces defining these projectors. Let $n$ be the order of the projectors $Q_{i}$, $r$ be their rank, then it is shown that the maximum number of linearly independent matrices $P_{i}=Q_{i}-Q_{0}$ such that the conditions $(Q_{i}-Q_{j})^{2}=0$ are satisfied is $r(n-r)$. Therefore, any projector of rank $r$ can be represented as the sum of an orthoprojector $Q_{0}$ and a linear combination of at most $r(n-r)$ projectors $Q_{i}$ so that $(Q_{i}-Q_{j})^{2}=0$, $i,j=0,1,\dots,r(n-r)$. The paper calculates the minimum distance between two projectors of ranks $k$ and $l - |k-l|^{1/2}$. The maximum distance between two orthoprojectors of the same rank $k$ is $(2k)^{1/2}$. It is established that the polynomial $h(p,q)=(p-q)^{2}$ plays a special role for the algebra $\mathcal {A}(p,q)$ generated by the projectors $p,q,I$. The polynomial $h$ generates the center of this algebra — the set of elements commuting with all elements of $\mathcal {A}(p,q)$.

Détail

Keywords: projector, linear manifold, linear subspace of matrices of bounded rank, block-triangular form pair of projectors, center of an algebra generated by two projectors.