Linear manifolds of projectors
https://doi.org/10.22405/2226-8383-2022-23-2-42-55
Abstract
The paper shows that a linear manifold of matrices of the form: Q=Q0+Σ︀a𝑖P𝑖, can consist of projectors only. It turns out that for this it is necessary and sufficient that P𝑖 =Q𝑖-Q0 and all the matrices Q𝑖 be projectors, moreover: (Q𝑖-Q𝑗)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 (AB) 2=0, one of them in terms of the subspaces defining these projectors.
Let n be the order of the projectors Q𝑖, r be their rank, then it is shown that the maximum number of linearly independent matrices P𝑖=Q𝑖-Q0 such that the conditions (Q𝑖-Q𝑗)2=0 are
satisfied is r(n-r). Therefore, any projector of rank r can be represented as the sum of an orthoprojector Q0 and a linear combination of at most r(n-r) projectors Q𝑖 so that (Q𝑖-Q𝑗)2=0,
i,j=0,1,..,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 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 𝒜(𝑝, 𝑞).
About the Author
Alexander Mikhailovich Vetoshkin
Moscow State Technical University N. E. Bauman (Mytishchi branch)
Russian Federation
Russian Federation
candidate of technical sciences, associate professor
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Review
For citations:
Vetoshkin A.M. Linear manifolds of projectors. Chebyshevskii Sbornik. 2022;23(2):42-55. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-2-42-55
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