2.5 Matrix With Cyclic Structure

Remark # of distinct peripheral eigenvalues of A = cyclic index of A When A is an irreducible nonnegative matrix = the index of imprimitivity of G(A) = spectral index of A = the largest m such that

sgn(τ) A cyclic permutation a product of transpositions.

sgn( π ) Any permutation can be writen of the form: where eachis cyclic permuation.

Remark If is n. If G(A) has no circuits, then detA=0, then there are (vertex-) disjoint circuits in G(A), the sum of lengths

E k (A) = the sum of nonzero terms of the (or form the sum of lengths is equal to k. where ) are disjoint circuits

Remark If G(A) has no circuits, then then i.e. A is nilpotent

Example G(A)

Example G(A)

Cofactor of A Let If

Example Consider by digraph 5 6

C sr of the form where αis a path from vertex r to vervex s In general,is the sum of possible terms are circuits s.t. the path and and or the circuits are mutually disjoint and together they contain all vertices of G(A)

D n complete digraph of order n circuit Consider as an edge-weighted digraph weight of

weight of D(π) permutation digraph product of weights of circuits

Example Let then

Remark (i) If A and B have the same set of circuits for each circuit Let and as a consequence and then A and B have equal corresponding principal minors of all possible orders

Fact cf. Exercise why does A must be irreducible? see next second page.

G(A)=G(B) is not irreducibole But A and B are not diagonally similar. does not appear in circuit product

Remark (ii) minors A and B have equal corresponding principal A and B have the same circuit products. principal minors A and B have the same corresponding A Counter example

Counter Example 1 G(A): G(B):

Counter Example 2 A and A T have the same principal minors But we may have

Question A and B have the same principal minors Hartfiel and Loewy proved the following:

Introduce A Semiring are associative,commutative R + form a semiring under On introduce by: distributes over 0 is zero element

max-product algebra A ⊕ B p.1

A ⊕ B p.2

Fuzzy Matrix Version max-min algebra

Boolean Matrix B: (0,1)-matrixdifferent from F 2 Spectial case of max-min algebra

In max-product algebra, max-min algebra satisfies the associative low.

in the sence max algebra or fuzzy algebra

a directed walk of length two in G(A) from i to j

a directed walk of length k in G(A) from i to j Furthermore, G(A) contain the directed walk the sum of walk products of A w.r.t the directed walks in G(A) from i to j of length k

In the setting of Max algebra (max-product algebra) p.1 a directed walk of length two in G(A) from i to j the maximum of walk products of A w.r.t the directed walks in G(A) from i to j of length two

In the setting of Max algebra (max-product algebra) p.2 a directed walk of length k in G(A) from i to j the maximum of walk products of A w.r.t the directed walks in G(A) from i to j of length k

In the setting of Fuzzy Matrix (max-min algebra) a directed walk of length k in G(A) from i to j It is difficult to explain the geometric meaning of

In the setting of Fuzzy Matrix (max-min algebra) p.2 a directed walk of length k in G(A) from i to j Furthermore, G(A) contain the directed walk It is difficult to explain the geometric meaning of from i to j of length k

Combinatorial Spectral Theory of Nonnegative Matrices

Plus max alge They are isometric

Remark p.2 space and Then for any Let X be a Topology space, F is a Banach is continuous map such that T(X) is precompact in F. there is a continuous mapof finite rank s.t.

sgn( π ) A permutation where eachis cyclic permuation.