# Distance between lattices of invariant subspaces of matrices

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.

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 $M$.