394 APPENDIX B. POSITIVE MATRICES

Proof: To verify the claim about the determinants, note(A 0

B C

)=

(A 0

0 I

)(I 0

B C

)

Therefore,

det

(A 0

B C

)= det

(A 0

0 I

)det

(I 0

B C

).

But it is clear from the method of Laplace expansion that

det

(A 0

0 I

)= detA

and from the multilinear properties of the determinant and row operations that

det

(I 0

B C

)= det

(I 0

0 C

)= detC.

The case where M is upper block triangular is similar.This immediately implies σ (M) = σ (A) ∪ σ (C) .

Theorem B.0.9 Let A > 0 and let λ0 be given in 2.7. If λ is an eigenvalue for A suchthat |λ| = λ0, then λ/λ0 is a root of unity. Thus (λ/λ0)

m= 1 for some m ∈ N.

Proof: Applying Theorem B.0.7 to AT , there exists v > 0 such that ATv = λ0v. Inthe first part of the argument it is assumed v >> 0. Now suppose Ax = λx,x ̸= 0 and that|λ| = λ0. Then

A |x| ≥ |λ| |x| = λ0 |x|

and it follows that if A |x| > |λ| |x| , then since v >> 0,

λ0 (v, |x|) < (v,A |x|) =(ATv, |x|

)= λ0 (v, |x|) ,

a contradiction. Therefore,A |x| = λ0 |x| . (2.8)

It follows that ∣∣∣∣∣∣∑j

Aijxj

∣∣∣∣∣∣ = λ0 |xi| =∑j

Aij |xj |

and so the complex numbers,Aijxj , Aikxk

must have the same argument for every k, j because equality holds in the triangle in-equality. Therefore, there exists a complex number, µi such that

Aijxj = µiAij |xj | (2.9)

and so, letting r ∈ N,Aijxjµ

rj = µiAij |xj |µr

j .

Summing on j yields ∑j

Aijxjµrj = µi

∑j

Aij |xj |µrj . (2.10)