8.2 Eigenvalues and Eigenvectors 109
MATLAB-Minute.
The roots of a polynomial p=αntn+αn−1tn−1+. . .+α0can be computed (or approximated) in MATLAB using the commandroots(p), wherepis a 1×(n+1)matrix with the entriesp(i)=αn+1−ifori=1, . . . ,n+1. Compute roots(p)for the monic polynomialp=t3−3t2+3t−1∈R[t]and display the output usingformat long. What are the exact roots of pand how large is the numerical error in the computation of the roots usingroots(p)?
Form the matrixA=compan(p)and compare its structure with the one of the companion matrix from Lemma8.4. Can you transfer the proof of Lemma8.4 to the structure of the matrixA?
Compute the eigenvalues of Awith the commandeig(A)and compare the output with the one ofroots(p). What do you observe?
A matrix with real entries is calledpositive, if all its entries are positive.
Lemma 8.15 If A ∈ Rn,n is positive and column-stochastic and if x ∈ Rn,1is an eigenvector of A corresponding to the eigenvalue 1, then either x or−x is positive.
Proof If x = [x1, . . . ,xn]T is an eigenvector of A = [ai j] corresponding to the eigenvalue 1, then
xi = n
j=1
ai jxj, i =1, . . . ,n.
Suppose that not all entries ofxare positive or not all entries ofxare negative. Then there exists at least one indexkwith
|xk|=
n j=1
ak jxj<
n j=1
ak j|xj|,
which implies n
i=1
|xi|<
n i=1
n j=1
ai j|xj| = n
j=1
n i=1
ai j|xj| = n
j=1
|xj| · n
i=1
ai j
=1
= n
j=1
|xj|.
This is impossible, so that indeedxor−xmust be positive. ⊓⊔ We can now prove the following uniqueness result.
Theorem 8.16 If A ∈ Rn,n is positive and column-stochastic, then there exists a unique positive x = [x1, . . . ,xn]T ∈Rn,1withn
i=1xi =1and Ax=x.
Proof By Lemma8.15, Ahas a least one positive eigenvector corresponding to the eigenvalue 1. Suppose that x(1) =
x1(1), . . . ,xn(1) T and x(2) =
x1(2), . . . ,xn(2) T are two such eigenvectors. Suppose that these are normalized by n
i=1xi(j) = 1, j = 1,2. This assumption can be made without loss of generality, since every nonzero multiple of an eigenvector is still an eigenvector.
We will show thatx(1)=x(2). Forα∈Rwe definex(α):=x(1)+αx(2)∈Rn,1, then
Ax(α)=Ax(1)+αAx(2)=x(1)+αx(2)=x(α).
If !α := −x1(1)/x1(2), then the first entry of x(!α) is equal to zero and thus, by Lemma8.15,x(!α)cannot be an eigenvector ofAcorresponding to the eigenvalue 1.
NowAx(!α)=x(!α)implies thatx(!α)=0, and hence
xi(1)+!αxi(2)=0, i =1, . . . ,n. (8.2)
8.3 Eigenvectors of Stochastic Matrices 111
Summing up thesenequations yields n
i=1
xi(1)
=1
+!α n
i=1
xi(2)
=1
=0,
so that!α = −1. From (8.2) we get xi(1) = x(2)i fori = 1, . . . ,n, and therefore
x(1)=x(2). ⊓⊔
The unique positive eigenvectorxin Theorem8.16is called thePerron eigenvec- tor2of the positive matrixA. The theory of eigenvalues and eigenvectors of positive (or more general nonnegative) matrices is an important area of Matrix Theory, since these matrices arise in many applications.
By construction, the matrix A ∈ Rn,n in the PageRank algorithm is column- stochastic but not positive, since there are (usually many) entriesai j =0. In order to obtain a uniquely solvable problem one can use the following trick:
Let S = [si j] ∈ Rn,n withsi j = 1/n. Obviously, S is positive and column- stochastic. For a real numberα∈(0,1]we define the matrix
A(α):=(1−α)A+αS.
This matrix is positive and column-stochastic, and hence it has a unique positive eigenvectorucorresponding to the eigenvalue 1. We thus have
u= A(α)u=(1−α)Au+αSu=(1−α)Au+α
n [1, . . . ,1]T.
For a very large number of documents (e.g. the entire internet) the numberα/n is very small, so that(1−α)Au≈u. Therefore a solution of the eigenvalue problem A(α)u =ufor smallαpotentially gives a good approximation of au ∈ Rn,1that satisfies Au =u. The practical solution of the eigenvalue problem with the matrix
A(α)is a topic of the field of Numerical Linear Algebra.
The matrixSrepresents a link structure where all document are mutually linked and thus all documents are equally important. The matrix A(α)=(1−α)A+αS therefore models the following internet “surfing behavior”: A user follows a proposed link with the probability 1−αand an arbitrary link with the probabilityα. Originally, Google Inc. used the valueα=0.15.
2Oskar Perron (1880–1975).
Exercises
(In the following exercisesK is an arbitrary field.)
8.1 Determine the characteristic polynomials of the following matrices overQ:
A= 2 0
0 2 , B = 4 4
−1 0 , C= 2 1
0 2 , D=
⎡
⎣ 2 0−1 0 2 0
−4 0 2
⎤
⎦.
Verify the Cayley-Hamilton theorem in each case by direct computation. Are two of the matricesA,B,Csimilar?
8.2 LetRbe a commutative ring with unit andn≥2.
(a) Show that for every A ∈G Ln(R)there exists a polynomial p ∈ R[t]of degree at mostn−1 with adj(A) = p(A). Conclude that A−1 =q(A) holds for a polynomialq∈ R[t]of degree at mostn−1.
(b) Let A ∈ Rn,n. Apply Theorem7.18 to the matrixt In − A ∈ (R[t])n,n and derive an alternative proof of the Cayley-Hamilton theorem from the formula det(t In−A)In =(t In−A)adj(t In−A).
8.3 Let A ∈ Kn,n be a matrix with Ak = 0 for somek ∈ N. (Such a matrix is callednilpotent.)
(a) Show thatλ=0 is the only eigenvalue of A.
(b) DeterminePA and show thatAn=0.
(Hint:You may assume thatPAhas the form
"n i=1
(t−λi)for someλ1, . . . ,λn
∈ K.)
(c) Show thatµIn−Ais invertible if and only ifµ∈K \ {0}.
(d) Show that(In−A)−1=In+A+A2+. . .+An−1.
8.4 Determine the eigenvalues and corresponding eigenvectors of the following matrices overR:
A=
⎡
⎣1 1 1 0 1 1 0 0 1
⎤
⎦, B=
⎡
⎣3 8 16
0 7 8
0−4−5
⎤
⎦, C =
⎡
⎢⎢
⎣
0−1 0 0
1 0 0 0
0 0−2 1 0 0 0−2
⎤
⎥⎥
⎦.
Is there any difference when you considerA,B,Cas matrices overC?
8.5 Letn ≥3 andε∈R. Consider the matrix
A(ε)=
⎡
⎢⎢
⎢⎢
⎣ 1 1
. .. . .. . ..1
ε 1
⎤
⎥⎥
⎥⎥
⎦
8.3 Eigenvectors of Stochastic Matrices 113 as an element ofCn,nand determine all eigenvalues in dependence ofε. How many pairwise distinct eigenvalues doesA(ε)have?
8.6 Determine the eigenvalues and corresponding eigenvectors of
A=
⎡
⎣2 2−a 2−a 0 4−a 2−a 0−4+2a −2+2a
⎤
⎦∈R3,3, B =
⎡
⎣1 1 0 1 0 1 0 1 1
⎤
⎦∈(Z/2Z)3,3.
(For simplicity, the elements ofZ/2Zare here denoted bykinstead of[k].) 8.7 Let A ∈ Kn,n, B ∈ Km,m,n ≥ m, and C ∈ Kn,m with rank(C) = mand
AC =C B. Show that then every eigenvalue ofBis an eigenvalue ofA.
8.8 Show the following assertions:
(a) trace(λA+µB)=λtrace(A)+µtrace(B)holds for allλ,µ∈ K and A,B∈ Kn,n.
(b) trace(A B)=trace(B A)holds for allA,B∈Kn,n. (c) IfA,B∈Kn,nare similar, then trace(A)=trace(B).
8.9 Prove or disprove the following statements:
(a) There exist matricesA,B∈Kn,nwith trace(A B)=trace(A)trace(B).
(b) There exist matricesA,B∈Kn,nwithA B−B A=In.
8.10 Suppose that the matrix A = [ai j] ∈ Cn,n has only real entries ai j. Show that ifλ ∈ C\Ris an eigenvalue of Awith corresponding eigenvectorv = [ν1, . . . ,νn]T ∈ Cn,1, then alsoλis an eigenvalue of A with corresponding eigenvectorv:= [ν1, . . . ,νn]T.
Vector Spaces
In the previous chapters we have focussed on matrices and their properties. We have defined algebraic operations with matrices and derived important concepts associ- ated with them, including their rank, determinant, characteristic polynomial, and eigenvalues. In this chapter we place these concepts in a more abstract framework by introducing the idea of a vector space. Matrices form one of the most important examples of vector spaces, and properties of certain (namely, finite dimensional) vector spaces can be studied in a transparent way using matrices. In the next chapter we will study (linear) maps between vector spaces, and there the connection with matrices will play a central role as well.