Statistical Physics 2, Fall 2015

Instructor: Yih-Yuh Chen

Exercise 2- on Nonnegative Matrices

No due date. For your entertainment only.

1. We did not actually cover the proof on the algebraic multiplicity of the largest eigenvalue of a positive

matrix in class. So here is a simple fix. Suppose its algebraic multiplicity were greater than one, then,

resorting to Jordan canonical form of linear algebra, we know that there would exist a vector 2 besides

the eigenvector 1 such that

(1) = 1

(2) = 2 + 1(1)

Thus, we have a two-dimensional invariant subspace spanned by {1 2}. Consider the action of on

the rays in this plane. The circle in the following figure represents all the rays of unit length. The

upper right red circular arc is the attracting “first quadrant” we talked about in class, while the lower

left green arc is antipodal to the red arc and is also attracting (why?).

(a) Consider a point on the arc 2 → and its image 0 ≡ ( ). When we let run from 2 to

we see that the sense of−−→ 0 gets reversed. Explain why this means that we must have another

eigenvector pointing at some angle between−−→2 and

−→?

(b) Why does this contradict Eqn. (1)? (We thus conclude that the algebraic multiplicity of must

be one.)

2. This exercise is meant to help you visualize how certain special positive matrices behave. Thus, consider

≡⎛⎝

⎞⎠

whose action on , and is cyclic. We assume , , and are all positive.

(a) Verify that the largest eigenvalue is + + and find its associated normalized eigenvector .

(b) Define 1 ≡ − | , 2 ≡ − | , 3 ≡ − | , which are orthogonal to

. (Note that the three ’s are not linearly independent!) Please explicitly verify that the linear

span of the three ’s is a two-dimensional invariant subspace of .

(c) Verify that (1) = 1 + 2 + 3, and cyclic relations.

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(d) Consider the following example:

≡⎛⎝

⎞⎠ ≡⎛⎝ 3 1 3

3 3 1

1 3 3

⎞⎠

please describe the action of on the vectors in .

3. Suppose we have discretized space and are trying to solve Laplace equation ∇2 = 0 in the followingtwo-dimensional shaded domain. It is assumed that is known for the red boundary points. Our

task is to find for the blue interior points, assuming the following discretized version of the Laplace

equation:

0 = ∇2 = 2

2+

2

2≈

³()+1 +

()−1 − 2()

´+³()+1 +

()−1 − 2()

´2

=

³()+1 +

()−1 +

()+1 +

()−1 − 4()

´2

=4

2·Ã()+1 +

()−1 +

()+1 +

()−1

4−

()

!

where is a small length. Assume there are blue interior points, and À 1, so that we can get a

good enough accuracy in the numerical solution.

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(a) Explain why this amounts to solving the following set of equations³ −

´ = −

where is the × identity matrix, is some × non-negative matrix with (very simple)

constant matrix coefficients, is an × 1 column vector whose entries are linearly related to theboundary values of , whereas is an ×1 column vector with entries consisted of the unknown ’s. In the above, the negative sign in front of

is there for convenience only.

(b) A very effective way of solving this problem is the relaxation method. Here is the idea. To solve³ −

´ = − (2)

one recasts the above into = +

and then sets up the following iteration scheme:

(+1)

= ()+

with an initial guess (0)of for all the blue interior points. Please convince yourself that this

means we are iteratively performing the update using the following scheme

(+1) ≡

()+1 +

()−1 +

()+1 +

()−1

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for all the blue interior points. That is, for the (+ 1)trial solution (+1) at the coordinate

( ), we assign it the value of the average of () of the surrounding points (the four dark blue

points) from the trial solution.

(c) Explain whyX=1

= 1

for the majority of , while for the rest of we have

X=1

1

(d) Ditto, but with the summation performed over the second index instead of the first index .

(e) Argue why eventually becomes a positive matrix for a large enough . (Hint: Investigate how

the value of an interior point gradually “diffuses” out to affect all the other interior points in

this scheme.)

(f) The above suggests that the largest eigenvalue of has multiplicity one. This turns out to be true,

and is what Perron-Frobenius theorem is all about. Let this eigenvalue and associated positive

eigenvector be denoted by and , respectively. Please use

=

=⇒X

=1=1

=

X=1

to show that 1.

(g) Show that

(+1)

=³ + −1 + +

´

= − +1

−

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(h) Explain why the scheme is effectively telling us that

=1

−

which indeed does agree with Eqn. (2) we started with!

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