Appendix B

Positive MatricesEarlier theorems about Markov matrices were presented. These were matrices in which allthe entries were nonnegative and either the columns or the rows added to 1. It turns outthat many of the theorems presented can be generalized to positive matrices. When this isdone, the resulting theory is mainly due to Perron and Frobenius. I will give an introductionto this theory here following Karlin and Taylor [19].

Definition B.0.1 For A a matrix or vector, the notation, A >> 0 will mean every entryof A is positive. By A > 0 is meant that every entry is nonnegative and at least one ispositive. By A ≥ 0 is meant that every entry is nonnegative. Thus the matrix or vectorconsisting only of zeros is ≥ 0. An expression like A >> B will mean A − B >> 0 withsimilar modifications for > and ≥.

For the sake of this section only, define the following for x =(x1, · · · , xn)T , a vector.

|x| ≡ (|x1| , · · · , |xn|)T .

Thus |x| is the vector which results by replacing each entry of x with its absolute value1.Also define for x ∈ Cn,

||x||1 ≡∑k

|xk| .

Lemma B.0.2 Let A >> 0 and let x > 0. Then Ax >> 0.

Proof: (Ax)i =∑

j Aijxj > 0 because all the Aij > 0 and at least one xj > 0.

Lemma B.0.3 Let A >> 0. Define

S ≡ {λ : Ax > λx for some x >> 0} ,

and letK ≡ {x ≥ 0 such that ||x||1 = 1} .

Now defineS1 ≡ {λ : Ax ≥ λx for some x ∈ K} .

Thensup (S) = sup (S1) .

Proof: Let λ ∈ S. Then there exists x >> 0 such that Ax > λx. Consider y ≡ x/ ||x||1 .Then ||y||1 = 1 and Ay > λy. Therefore, λ ∈ S1 and so S ⊆ S1. Therefore, sup (S) ≤sup (S1) .

Now let λ ∈ S1. Then there exists x ≥ 0 such that ||x||1 = 1 so x > 0 and Ax > λx.Letting y ≡ Ax, it follows from Lemma B.0.2 that Ay >> λy and y >> 0. Thus λ ∈ Sand so S1 ⊆ S which shows that sup (S1) ≤ sup (S) . ■

This lemma is significant because the set, {x ≥ 0 such that ||x||1 = 1} ≡ K is a compactset in Rn. Define

λ0 ≡ sup (S) = sup (S1) . (2.1)

The following theorem is due to Perron.

1This notation is just about the most abominable thing imaginable because it is the same notation butentirely different meaning than the norm. However, it saves space in the presentation of this theory ofpositive matrices and avoids the use of new symbols. Please forget about it when you leave this section.

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