Three different ways There are three different ways to show that ρ(A) is a simple eigenvalue of an irreducible nonnegative matrix A:

Exercise Suppose that If A is irreducible and A ≠B, then

Remark such that with equality iff the nonzero w i s have the argument i.e.

Lemma (Wielandt’s Lemma) p.1 Let Suppose that A is irreducible, nonnegative, andThen (a)

Lemma (Wielandt’s Lemma) p.2 (b) Suppose that C has an eigenvalue then C is of the form where D is a diagonal matrix that satisfies

Exercise p.1 and let k be a positive integer. (i) Prove that where the equality holds if and only if

Exercise p.2 the algebraic multiplicity of (i) Prove also that when the equality holds, algebraic multiplicity of as an eigenvalue of A is equal to the as an eigenvalue of A.

Exercise p.3 and Perron’s theorem to deduce that if A (ii) Use the result of part (i), Theorem is a simple eigenvalue of A. is an irreducible nonnegative matrix, then

Exercise p.1 Prove that if(i) Let and if A is irreducible Hint: Use Remark then

Exercise p.2 if A is an irreducible nonnegative matrix, (ii) Use the result of part (i) to deduce that different from A. for any principal submatrix B of A, then

Exercise p.1 relation where y,z are Perron vectors of A and Letbe an irreducible nonnegative matrix. (i) Let B(t) =adj(tI-A). By using the multiplicity of ρ(A) as an eigenvalue of A is 1, at t=ρ(A) and the fact that the geometric show that B(ρ(A)) is of the form respectively, and c is a nonzero real constant.

Exercise p.2 is the (n-1)-square principal the result of part (i) to deduce that (ii)Show that and column. (These relations are true for a where and hence submatrix obtained from A by deleting its row general square complex matrix A.) Then use is a simple eigenvalue of A.

Exercise p.3 the largest real root of (iii) Using the fact that cannot be negative. Deduce that is equal to explain why

Exercise matrix. Let B(t) have the same meaning as Let Exercise (ii) and the relation be an irreducible nonnegative given in Exercise Use the result of (which was established) to show that

Exercise p.1 Let A be an irreducible nonnegative matrix with index of imprimitivity h (i) Prove that trace(A)=0 whenever h>1

Exercise p.2 Let A be an irreducible nonnegative matrix with index of imprimitivity h (ii) Prove that h is a divisor of the number of nonzero eigenvalues of A (counting algebraic multiplicities)

Exercise p.3 Let A be an irreducible nonnegative matrix with index of imprimitivity h (iii) Prove that if A is nonsingular, nxn and n is a prime, then h=1 or n.

Theorem (2 nd part of the Frobenius Thm) Given irreducible matrix with m distinct eigenvalues with moduloρ(A) Then (i) the peripheral spectrum of A is

Theorem (2 nd part of the Frobenius Thm) (ii) (iii) If Apply Wielandt’s Lemma to prove

Another proof in next page

Fully Cyclic The peripheral spectrum of A is called fully cyclic if then is an eigenvector of A corresponding to for all integers k.

Fully Cyclic p.2 Note that for k=0, the latter condition becomes

Corollary Let A be an irreducible nonnegative matrix then the index of imprimitivity of A is equal to the spectral index of A.

Corollary The peripheral spectrum of an irreducible nonnegative matrix is fully cyclic

Exercise that the peripheral spectrum of P is fully If Let P be a square nonnegative matrix. Prove cyclic if and only if P satisfies the following: z is a corresponding eigenvector, then is a peripheral eigenvalue of P and

Theorem If then there is a permutation matrix P such that whereis 1x1 or is irreducible. Frobenius normal form

Example

V 1 V 2 V 3 Three components all are maximal Strongly connected component

Remark p.1 If A is reducible, nonnegative, and there is a permutation matrix P such that whereis 1x1 or is irreducible.

Remark p.2 then form of The peripheral eigenvalue of A is of the times a root of unity and

Theorem Given A is primitive if and only if

Exercise The spectral radius of a nonnegative matrix A is positive if and only if G(A) contains at least a circuit.

Exercise (i) Find the Frobenius normal form of the matrix (ii) Compute

Exercise matrix and let vector x such that Let A be an nxn irreducible nonnegative and a positive Then there exists where is defined to be

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

A ⊕ B p.1

A ⊕ B p.2

Circuit Geometric mean circuit is called circuit geometric mean

Lemma Let A be an irreducible nonnegative matrix x is a semipositive vector and such that then where μ(A) is the maximum circuit geometric mean. γ is called a max eigenvalue and x is the corresponding max eigenvector

Plus max algebra R ∪ {-∞} They are isometric

Brouwer’s Fixed Point Theorem Let C be a compact convex subset of and f is a continuous map from C to C, then

Theorem Max version of the Perron-Frobenius Theorem If A be an irreducible nonnegative matrix then there is a positive vector x such that

Fact diagonal entries such that has the constant row sums. there is a diagonal matrix D with positive Given an irreducible nonnegative matrix A

Remark diagonal entries such that in Theorem is equivalent to the assertion there is a diagonal matrix D with positive that given an irreducible nonnegative matrix A the maximum entry in each row is the same.

If Let k be a fixed positive integer, the sum of the k largest, then let us define components in then

If Let k be a fixed positive integer, by, then let us define then

Theorem and let constant Let A be an irreducible nonnegative matrix Then there exists a positive vector x and a such that

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.

Combinatorial Spectral Theory of Nonnegative Matrices