行列はグラフであり、グラフは行列である
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行列はグラフであり、グラフは行列である
@TivadarDanka: The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices. Encoding matrices as graphs is a cheat code, making complex behavior simple to study.
Let me show you how!
https://pbs.twimg.com/media/FmBy7vEacAALjBw.jpg
https://gyazo.com/15e311f6432a827403aa95100c56abb6
画像の例だと、緑の点が持ってる向きの力を示している
この関係が自分から存在しない場合、0が入る
https://gyazo.com/54af21ab4837c0d274fe70649e33936e
https://gyazo.com/3e3b242afa3cd9d5e9c7e2281f94e96e
よりわかりやすくする
@TivadarDanka: Why is the directed graph representation beneficial for us? For one, the powers of the matrix correspond to walks in the graph.
Take a look at the elements of the square matrix. All possible 2-step walks are accounted for in the sum defining the elements of A².
https://pbs.twimg.com/media/FmBy-GqaAAAimSm.jpg
ここら辺から全くわからなくなる
@TivadarDanka: A directed graph is strongly connected if every node can be reached from every other node. If this is not true, the graph is not strongly connected.
Below, you can see an example of both.
https://pbs.twimg.com/media/FmBy_EBacAASndO.jpg
@TivadarDanka: The corresponding matrix of our graph can be reduced to a simpler form! Its diagonal comprises blocks whose graphs are strongly connected. (That is, the blocks are irreducible.) Furthermore, the block below the diagonal is zero.
https://pbs.twimg.com/media/FmBzBcmacAATh9T.jpg
@TivadarDanka: In general, this block-matrix structure is called the Frobenius normal form. https://pbs.twimg.com/media/FmBzCKZacAAl154.jpg