Hi, I went through few linear algebra books and I have few questions about eigenvalues.
1. The characteristic polynomial det|A-eI| = 0 can be solved to get eigenvalues e1, e2....
I understand that det|A-eI| has to be 0 so that the matrix 'A-eI' is not invertible. But looking at this equation geometrically, it means that volume of structure formed by column vectors of 'A-eI' has to be zero. Can anyone explain how the volume enclosed by column vectors and eigenvalues are related to each other.
2. Can any one explain eigenvalues in terms of graph structure. For example : We can say that determinant is determined by number and even/odd cycles in graphs. Is there any such relation between eigen values and graph.