This connection really clicks for me. The matrix-graph duality definately makes eigenvalues way more intuitive when you think about them as describing graph structure. I worked on some network analysis stuff last year and wish I'd seen this framing then—would've saved alot of debugging time. Though I wonder if this perspective gets harder with really sparse matrices where the graph becomes almost trivial.
If matrices are graphs, and graphs are tropical curves, then are matrices also tropical curves? Also tropical curves relate to non-archemedian geometry. Is this latter form of geometry, which is like a geometry of hierarchical tress good for biological data?
This connection really clicks for me. The matrix-graph duality definately makes eigenvalues way more intuitive when you think about them as describing graph structure. I worked on some network analysis stuff last year and wish I'd seen this framing then—would've saved alot of debugging time. Though I wonder if this perspective gets harder with really sparse matrices where the graph becomes almost trivial.
Between 7:20 and 7:30 the last equation is missing the transpose notation — it currently shows PAP but it should P(t) A P
Please confirm if my observation is correct
Thanks
Hey! The transposition matrices are invariant to matrix transposition, so the last line is correct!
Got it, thanks for clarifying!
If matrices are graphs, and graphs are tropical curves, then are matrices also tropical curves? Also tropical curves relate to non-archemedian geometry. Is this latter form of geometry, which is like a geometry of hierarchical tress good for biological data?