Question

For a linear transformation $A: C^n \to C^n$ let $Inv(A)$ be the lattice of all $A$-invariant subspaces. In work I.~Gohberg, L.~Rodman "On the Distance between Lattices of Invariant Subspaces of Matrices" analysis the distanse between $Inv(A)$ and $Inv(B)$ defined as follows: $$ dist(Inv(A),Inv(B))=\max\Big(\sup\limits_{M\in Inv(A)}\ \ \inf\limits_{L\in Inv(B)}||P_L-P_M||, \sup\limits_{M\in Inv(B)}\ \ \inf\limits_{L\in Inv(A)}||P_L-P_M||\Big), $$ where $P_N$ is orthogonal projector on the subspace $N$ in $C^n$.

I can not quite understand this definition. Why is there symmetry? Maybe someone met geometric Interpretation of these definition.

Thanks for your help.

Answer 1

Essentially this is nothing else than the Hausdorff distance between these lattices (as sets of subspaces) if you consider $\|P_L-P_M\|$ as the distance between subspaces $L$ and $M$.