Representation theory: example, isomorphisms and homomorphisms
... primitive). Then G has a 1-dimensional representation given by ρ(x) = ζ x . Example 5. Let G act on a finite set X. Then there is an associated representation on FX , the vector space with basis X, given by ...
... primitive). Then G has a 1-dimensional representation given by ρ(x) = ζ x . Example 5. Let G act on a finite set X. Then there is an associated representation on FX , the vector space with basis X, given by ...
Chapter 8: Markov Chains
... These results are summarized in the following Theorem. Theorem 8.7: Let {X0, X1 , X2, . . .} be a Markov chain with N × N transition matrix P . If the probability distribution of X0 is given by the 1 × N row vector π T , then the probability distribution of Xt is given by the 1 × N row vector π T P ...
... These results are summarized in the following Theorem. Theorem 8.7: Let {X0, X1 , X2, . . .} be a Markov chain with N × N transition matrix P . If the probability distribution of X0 is given by the 1 × N row vector π T , then the probability distribution of Xt is given by the 1 × N row vector π T P ...
The Adjacency Matrices of Complete and Nutful Graphs
... For a graph G = Γ(G), the vertices can be partitioned into G–core and G–coreforbidden vertices for G. A vertex of G corresponding to a non–zero entry in some kernel eigenvector x is said to be a G–core vertex (G–CV) for the adjacency matrix G associated with G. A G–core-forbidden vertex (G–CFV), for ...
... For a graph G = Γ(G), the vertices can be partitioned into G–core and G–coreforbidden vertices for G. A vertex of G corresponding to a non–zero entry in some kernel eigenvector x is said to be a G–core vertex (G–CV) for the adjacency matrix G associated with G. A G–core-forbidden vertex (G–CFV), for ...
Chapter 1 Notes
... 1. All rows consisting entirely of zeros are grouped at the bottom. 2. The leftmost nonzero number in each row is 1 (called the leading one). 3. The leading 1 of a row is to the right of the previous row's leading 1. 4. All entries directly above and below a leading 1 are zeros. ...
... 1. All rows consisting entirely of zeros are grouped at the bottom. 2. The leftmost nonzero number in each row is 1 (called the leading one). 3. The leading 1 of a row is to the right of the previous row's leading 1. 4. All entries directly above and below a leading 1 are zeros. ...
