## Part II

## On the Numerical Solutions of

## Eigenvalue Problems

## Chapter 5

## The Unsymmetric Eigenvalue Problem

Generalized eigenvalue problem (GEVP):

*Given A, B ∈ C*^{n×n}*. Determine λ ∈ C and 0 6= x ∈ C*^{n}*with Ax = λBx. λ is*
*called an eigenvalue of the pencil A − λB (or pair(A, B)) and x is called an eigen-*
*vector corresponding to λ. λ is an eigenvalue of A − λB ⇐⇒ det(A − λB) = 0.*

*(σ(A, B) ≡ {λ ∈ C | det(A − λB) = 0}.)*

*Definition 5.0.2 A pencil A − λB (A, B ∈ R*^{m×n}*) or a pair(A, B) is called regular if*
*that*

*(i) A and B are square matrices of order n, and*
*(ii) det(A − λB) 6≡ 0.*

*In all other case (m 6= n or m = n but det(A − λB) ≡ 0), the pencil is called singular.*

*Detailed algebraic structure of a pencil A − λB see Matrix theory II, chapter XII (Gant-*
macher 1959).

Eigenvalue Problem (EVP):

*Special case in GEVP when B = I, we have λ ∈ C and 0 6= x ∈ C*^{n}*with Ax = λx. λ is*
*an eigenvalue of A and x is an eigenvector corresponding to λ.*

*Definition 5.0.3 (a) σ(A) = {λ ∈ C | det(A − λI) = 0} is called the spectrum of A.*

*(b) ρ(A) = max{| λ |: λ ∈ σ(A)} is called the radius of σ(A).*

*(c) P (λ) = det(λI − A) is called the characteristic polynomial of A.*

*Let P (λ) =*
Y*s*
*i=1*

*(λ − λ** _{i}*)

^{m(λ}

^{i}^{)}

*,*

*λ*

_{i}*6= λ*

_{j}*(i 6= j) and*X

*s*

*i=1*

*m(λ*_{i}*) = n.*

*Example 5.0.2 A =*

· 2 2 0 3

¸
*, B =*

· 1 0 0 0

¸

*=⇒ det(A − λB) = 2 − λ and σ(A, B) =*
*{2}.*

*Example 5.0.3 A =*

· 1 2 0 3

¸
*, B =*

· 0 1 0 0

¸

*=⇒ det(A − λB) = 3 and σ(A, B) = ∅.*

*Example 5.0.4 A =*

· 1 2 0 0

¸
*, B =*

· 1 0 0 0

¸

*=⇒ det(A − λB) = 0 and σ(A, B) = C.*

*Example 5.0.5 det(µA − λB) = (2µ − λ)µ*
*µ = 1 : Ax = λBx =⇒ λ = 2.*

*λ = 1 : Bx = µAx =⇒ µ = 0, µ =* ^{1}_{2} *=⇒ λ = ∞, λ = 2.*

*σ(A, B) = {2, ∞}.*

*Example 5.0.6 det(µA − λB) = µ · 3µ*
*µ = 1 : no solution for λ.*

*λ = 1 : Bx = µAx =⇒ µ = 0, 0.(multiple)*
*σ(A, B) = {∞, ∞}.*

Let

*m(λ**i**) := algebraic multiplicity of λ**i*.

*n(λ*_{i}*) := n − rank(A − λ*_{i}*I) = geometric multiplicity.*

*1 ≤ n(λ*_{i}*) ≤ m(λ*_{i}*).*

*If for some i, n(λ**i**) < m(λ**i**), then A is degenerated (defective). The following statements*
are equivalent:

*(a) A is diagonalizable: There exists a nonsingular matrix T such that T*^{−1}*AT =*
*diag(λ*_{1}*, · · · , λ*_{n}*).*

*(b) There are n linearly independent eigenvectors.*

*(c) A is nondefective, i.e. ∀ λ ∈ σ(A) =⇒ m(λ) = n(λ).*

*If A is defective then eigenvector + principle vector =⇒ Jordan form.*

*Theorem 5.0.3 (Jordan decomposition) If A ∈ C*^{n×n}*, then there exists a nonsingu-*
*lar X ∈ C*^{n×n}*, such that X*^{−1}*AX = diag(J*_{1}*, · · · , J*_{t}*),where*

*J** _{i}* =

*λ** _{i}* 1 0

*. .. ...*

*. .. 1*

0 *λ*_{i}

*is m**i**× m**i* *and m*1*+ · · · + m**t**= n.*

*Theorem 5.0.4 (Schur decomposition) If A ∈ C*^{n×n}*then there exists a unitary ma-*
*trix U ∈ C*^{n×n}*such that U*^{∗}*AU(= U*^{−1}*AU) a upper triangular.*

*- A normal(i.e. AA*^{∗}*= A*^{∗}*A) ⇐⇒ ∃ unitary U such that U*^{∗}*AU = diag(λ*_{1}*, · · · , λ*_{n}*), i.e.*

*Au**i* *= λ**i**u**i**,* *u*^{∗}_{i}*u**j* *= δ**ij**.*

*- A hermitian(i.e. A*^{∗}*= A) ⇐⇒ A is normal and σ(A) ⊆ R.*

*- A symmetric(i.e. A*^{T}*= A, A ∈ R*^{n×n}*) ⇐⇒ ∃ orthogonal U such that U*^{T}*AU =*
*diag(λ*_{1}*, · · · , λ*_{n}*) and σ(A) ⊆ R.*

### 5.1 Orthogonal Projections and C-S Decomposition

*Definition 5.1.1 Let S ⊆ R*^{n}*be a subspace, P ∈ R*^{n×n}*is the orthogonal projection onto*
*S if*

*Range(P ) = S,*
*P*^{2} *= P,*

*P*^{T}*= P,*

(5.1.1)

*where Range(P ) = R(P ) = {y ∈ R*^{n}*| y = P x, for some x ∈ R*^{n}*}.*

*Remark 5.1.1 If x ∈ R*^{n}*, then P x ∈ S and (I − P )x ∈ S*^{⊥}*.*

*Example 5.1.1 P = vv*^{T}*/v*^{T}*v is the orthogonal projection onto S = span{v}, v ∈ R*^{n}*.*
x

Px

S=span{v}

Figure 5.1: Orthogonal projection

*Remark 5.1.2 (i) If P*_{1} *and P*_{2} *are orthogonal projections, then for any z ∈ R*^{n}*we have*
*k (P*_{1}*− P*_{2}*)z k*^{2}_{2}*= (P*_{1}*z)*^{T}*(I − P*_{2}*)z + (P*_{2}*z)*^{T}*(I − P*_{1}*)z.* (5.1.2)
*If R(P*_{1}*) = R(P*_{2}*) = S then the right-hand side of (5.1.2) is zero. Thus the orthog-*
*onal projection for a subspace is unique.*

*(ii) If V = [v*_{1}*, · · · , v*_{k}*] is an orthogonal basis for S, then P = V V*^{T}*is unique orthogonal*
*projection onto S.*

*Definition 5.1.2 Suppose S*_{1} *and S*_{2} *are subspaces of R*^{n}*and dim(S*_{1}*) = dim(S*_{2}*). We*
*define the distance between S*_{1} *and S*_{2} *by*

*dist(S*_{1}*, S*_{2}*) =k P*_{1}*− P*_{2} *k*_{2}*,* (5.1.3)
*where P*_{i}*is the orthogonal projection onto S*_{i}*,* *i = 1, 2.*

*Remark 5.1.3 By considering the case of one-dimensional subspaces in R*^{2}*, we obtain*
*a geometrical interpretation of dist(·, ·). Suppose S*1 *= span{x} and S*2 *= span{y} and*

S

S

=span{y}

=span{x}

2

θ

1

*k x k*_{2}*=k y k*_{2}*= 1. Assume that x*^{T}*y = cos θ, θ ∈ [0,*^{π}_{2}*]. It follows that the difference*
*between the projections onto these spaces satisfies*

*P*1*− P*2 *= xx*^{T}*− yy*^{T}*= x[x − (y*^{T}*x)y]*^{T}*− [y − (x*^{T}*y)x]y*^{T}*.*
*If θ = 0(⇒ x = y), then dist(S*_{1}*, S*_{2}*) =k P*_{1}*− P*_{2} *k*_{2}*= sin θ = 0.*

*If θ 6= 0, then*

*U*_{x}*= [u*_{1}*, u*_{2}*] = [x, −[y − (y*^{T}*x)x]/ sin θ]*

*and*

*V**x* *= [v*1*, v*2*] = [[x − (x*^{T}*y)y]/ sin θ, y]*

*are defined and orthogonal. It follows that*

*P*1*− P*2 *= U**x* *diag[sin θ, sin θ] V*_{x}^{T}

*is the SVD of P*_{1} *− P*_{2}*. Consequently, dist(S*_{1}*, S*_{2}*) = sin θ, the sine of the angle between*
*the two subspaces.*

Theorem 5.1.1 (C-S Decomposition, Davis / Kahan(1970) or Stewart(1977))
*If Q =*

· *Q*_{11} *Q*_{12}
*Q*_{21} *Q*_{22}

¸

*is orthogonal with Q*_{11} *∈ R*^{k×k}*and Q*_{22} *∈ R*^{j×j}*(k ≥ j), then there*
*exists orthogonal matrices U*_{1}*, V*_{1} *∈ R*^{k×k}*and orthogonal matrices U*_{2}*, V*_{2} *∈ R*^{j×j}*such that*

· *U*_{1} 0
0 *U*2

¸* _{T}* ·

*Q*_{11} *Q*_{12}
*Q*21 *Q*22

¸ · *V*_{1} 0
0 *V*2

¸

=

*I* 0 0

0 *C* *S*

*0 −S C*

* ,* (5.1.4)

*where*

*C = diag(c*1*, · · · , c**j**),* *c**i* *= cos θ**i**,*
*S = diag(s*_{1}*, · · · , s*_{j}*),* *s*_{i}*= sin θ*_{i}

*and 0 ≤ θ*_{1} *≤ θ*_{2} *≤ · · · ≤ θ*_{j}*≤* ^{π}_{2}*.*
*Lemma 5.1.1 Let Q =*

· *Q*_{1}
*Q*_{2}

¸

*be orthogonal with Q*_{1} *∈ R*^{n×n}*. Then there are unitary*
*matrices U*_{1}*, U*_{2} *and W such that*

· *U*_{1}* ^{T}* 0
0

*U*

_{2}

^{T}¸ · *Q*_{1}
*Q*_{2}

¸
*W =*

· *C*
*S*

¸

*where C = diag(c*_{1}*, · · · , c*_{j}*) ≥ 0, and S = diag(s*_{1}*, · · · , s*_{n}*) ≥ 0 with c*^{2}_{i}*+ s*^{2}_{i}*= 1, i =*
*1, · · · , n.*

*Proof: Let U*_{1}^{T}*Q*_{1}*W = C be the SVD of Q*_{1}. Consider

· *U*_{1}* ^{T}* 0
0

*I*

¸ · *Q*_{1}
*Q*_{2}

¸
*W =*

· *C*

*Q*_{2}*W*

¸

has orthogonal columns. Define ˜*Q*2 *≡ Q*2*W . Then C*^{2} + ˜*Q*^{T}_{2}*Q*˜2 *= I or ˜Q*^{T}_{2}*Q*˜2 *= I − C*^{2}
diagonal, thus ˜*Q*^{T}_{2}*Q*˜_{2} is diagonal. Which means that the nonzero column of ˜*Q*_{2} are
orthogonal to one another.If all the columns of ˜*Q*_{2} *are nonzero, set S*^{2} = ˜*Q*^{T}_{2}*Q*˜_{2} and
*U*2 = ˜*Q*2*S*^{−1}*, then we have U*_{2}^{T}*U*2 *= I and U*_{2}^{T}*Q*˜2 *= S. It follows the decomposition.*

If ˜*Q*_{2} has zero columns, normalize the nonzero columns and replace the zero columns
with an orthogonal basis for the orthogonal complement of the column space of ˜*Q*_{2}. It is
*easily verified that U*_{2} *so defined is orthogonal and S ≡ U*_{2}^{T}*Q*˜_{2} is diagonal. It also follows
that decomposition.

*Theorem 5.1.2 (C-S Decomposition) Let the unitary matrix W ∈ C*^{n×n}*be parti-*
*tioned in the form W =*

· *W*_{11} *W*_{12}
*W*_{21} *W*_{22}

¸

*, where W*_{11}*∈ C*^{r×r}*with r ≤* ^{n}_{2}*. Then there exist*

*unitary matrices U = diag(*

z}|{*r*

*U*_{1} *,*
z}|{*n−r*

*U*_{2} *) and V = diag(*

z}|{*r*

*V*_{1} *,*
z}|{*n−r*

*V*_{2} *) such that*

*U*^{∗}*W V =*

*Γ −Σ 0*

Σ Γ 0

0 0 *I*

*}r*
*}r*
*}n − 2r*

*,* (5.1.5)

*where Γ = diag(γ*_{1}*, · · · , γ*_{r}*) ≥ 0 and Σ = diag(σ*_{1}*, · · · , σ*_{r}*) ≥ 0 with γ*_{i}^{2} *+ σ*^{2}_{i}*= 1, i =*
*1, · · · , r.*

*Proof: Let Γ = U*_{1}^{∗}*W*_{11}*V*_{1} *be the SVD of W*_{11} *with the diagonal elements of Γ : γ*_{1} *≤*
*γ*_{2} *≤ · · · ≤ γ*_{k}*< 1 = γ*_{k+1}*= · · · = γ*_{r}*, i.e.*

*Γ = diag(Γ*^{0}*, I** _{r−k}*).

The matrix

· *W*_{11}
*W*_{21}

¸

*V*_{1} has orthogonal columns. Hence

*I =*

·µ *W*_{11}
*W*_{21}

¶
*V*_{1}

¸* _{∗}* ·µ

*W*

_{11}

*W*

_{21}

¶
*V*_{1}

¸

= Γ^{2}*+ (W*_{21}*V*_{1})^{∗}*(W*_{21}*V*_{1}*).*

*Since I and Γ*^{2} *are diagonal, (W*_{21}*V*_{1})^{∗}*(W*_{21}*V*_{1}*) is diagonal. So the columns of W*_{21}*V*_{1} are
*orthogonal. Since the ith diagonal of I − Γ*^{2} *is the norm of the ith column of W*_{21}*V*_{1}, only
*the first k(k ≤ r ≤ n − r) columns of W*_{21}*V*_{1} are nonzero. Let ˆ*U*_{2} be unitary whose first
*k columns are the normalized columns of W*21*V*1. Then

*U*ˆ_{2}^{∗}*W*_{21}*V*_{1} =

· Σ 0

¸
*,*

*where Σ = diag(σ*_{1}*, · · · , σ*_{k}*, 0, · · · , 0) ≡ diag(Σ*^{0}*, 0), ˆU*_{2} *∈ C**(n−r)×(n−r)*. Since

*diag(U*_{1}*, ˆU*_{2})^{∗}

µ *W*_{11}
*W*21

¶
*V*_{1} =

Γ Σ 0

*has orthogonal (orthonormal) columns, we have γ*_{i}^{2}*+ σ*_{i}^{2} *= 1, i = 1, · · · , r. (Σ** ^{0}* is nonsin-
gular).

*By the same argument as above : there is a unitary V*_{2} *∈ C**(n−r)×(n−r)* such that
*U*_{1}^{∗}*W*12*V*2 *= (T, 0),*

*where T = diag(τ*_{1}*, · · · , τ*_{r}*) and τ*_{i}*≤ 0. Since γ*_{i}^{2} *+ τ*_{i}^{2} *= 1, it follows from γ*_{i}^{2} *+ σ*_{i}^{2} = 1
*that T = −Σ. Set ˆU = diag(U*1*, ˆU*2*) and V = diag(V*1*, V*2*). Then X = ˆU*^{∗}*W V can be*
partitioned in the form

*X =*

Γ^{0}*0 −Σ** ^{0}* 0 0

0 *I* 0 0 0

Σ^{0}*0 X*33 *X*34 *X*35

0 *0 X*_{43} *X*_{44} *X*_{45}
0 *0 X*_{53} *X*_{54} *X*_{55}

*}k*
*}r − k*
*}k*
*}r − k*
*}n − 2r*

*.*

Since columns 1 and 4 are orthogonal, it follows Σ^{0}*X*_{34} *= 0. Thus X*_{34} = 0 (since Σ^{0}*nonsigular). Likewise X*_{35}*, X*_{43}*, X*_{53} = 0. From the orthogonality of columns 1 and 3, it
*follows that −Γ** ^{0}*Σ

*+ Σ*

^{0}

^{0}*X*

_{33}

*= 0, so X*

_{33}= Γ

*. The matrix ˆ*

^{0}*U*

_{3}=

· *X*_{44} *X*_{45}
*X*54 *X*55

¸

is unitary.

*Set U*2 *= diag(I**k**, ˆU*3) ˆ*U*2 *and U = diag(U*1*, U*2*). Then U*^{H}*W V = diag(I**r+k**, ˆU*3*)X with*

*X =*

Γ^{0}*0 −Σ** ^{0}* 0 0

0 *I* 0 0 0

Σ* ^{0}* 0 Γ

*0 0*

^{0}0 0 0 *I 0*

0 0 0 *0 I*

*.*

The theorem is proved.

*Theorem 5.1.3 Let W = [W*_{1}*, W*_{2}*] and Z = [Z*_{1}*, Z*_{2}*] be orthogonal, where W*_{1}*, Z*_{1} *∈ R*^{n×k}*and W*_{2}*, Z*_{2} *∈ R*^{n×(n−k)}*. If S*_{1} *= R(W*_{1}*) and S*_{2} *= R(Z*_{1}*) then*

*dist(S*_{1}*, S*_{2}) =
q

*1 − σ*_{min}^{2} *(W*_{1}^{T}*Z*_{1}) (5.1.6)
*Proof: Let Q = W*^{T}*Z and assume that k ≥ j = n − k. Let the C-S decomposition of Q*
*be given by (5.1.2), (Q*_{ij}*= W*_{i}^{T}*Z*_{j}*,* *i, j = 1, 2). It follows that*

*k W*_{1}^{T}*Z*2 *k*2*=k W*_{2}^{T}*Z*1 *k*2*= s**j* =
q

*1 − c*^{2}* _{j}* =p

*1 − σ*_{min}^{2} *(W*_{1}^{T}*Z*1).

*Since W*_{1}*W*_{1}^{T}*and Z*_{1}*Z*_{1}^{T}*are the orthogonal projections onto S*_{1} *and S*_{2}, respectively. We
have

*dist(S*_{1}*, S*_{2}*) = k W*_{1}*W*_{1}^{T}*− Z*_{1}*Z*_{1}^{T}*k*_{2}

*= k W*^{T}*(W*_{1}*W*_{1}^{T}*− Z*_{1}*Z*_{1}^{T}*)Z k*_{2}

*= k*

· 0 *W*_{1}^{T}*Z*_{2}
*W*_{2}^{T}*Z*1 0

¸
*k*_{2}

*= s*_{j}*.*

*If k < j, the above argument by setting Q = [W*_{2}*, W*_{1}]^{T}*[Z*_{2}*, Z*_{1}] and noting that
*σ**min**(W*_{2}^{T}*Z*1*) = σ**min**(W*_{1}^{T}*Z*2*) = s**j**.*

### 5.2 Perturbation Theory

*Theorem 5.2.1 (Gerschgorin Circle Theorem) If X*^{−1}*AX = D+F , D ≡ diag(d*_{1}*, · · · , d** _{n}*)

*and F has zero diagonal entries, then σ(A) ⊂*S

_{n}*i=1**D*_{i}*, where*
*D*_{i}*= {z ∈ C | |z − d*_{i}*| ≤*

X*n*
*j=1,j6=i*

*|f*_{ij}*|}.*

*Proof: Suppose λ ∈ σ(A) and assume without loss of generality that λ 6= d*_{i}*for i =*
*1, · · · , n. Since (D − λI) + F is singular, it follows that*

*1 ≤k (D − λI)*^{−1}*F k** _{∞}*=
X

*n*

*j=1*

*|f*_{kj}*| / |d*_{k}*− λ|*

*for some k(1 ≤ k ≤ n). But this implies that λ ∈ D** _{k}*.

*Corollary 5.2.1 If the union M*

_{1}=S

_{k}*j=1**D*_{i}_{j}*of k discs D*_{i}_{j}*, j = 1, · · · , k, and the union*
*M*2 *of the remaining discs are disjoint, then M*1 *contains exactly k eigenvalues of A and*
*M*_{2} *exactly n − k eigenvalues.*

*Proof: Let B = X*^{−1}*AX = D + F , for t ∈ [0, 1]. Let B*_{t}*:= D + tF , then B*_{0} =
*D, B*_{1} *= B. The eigenvalues of B*_{t}*are continuous functions of t. Applying Theorem 5.2.1*
*of Gerschgorin to B*_{t}*, one finds that for t = 0, there are exactly k eigenvalues of B*_{0} *in M*_{1}
*and n − k in M*_{2}*. (Counting multiple eigenvalues) Since for 0 ≤ t ≤ 1 all eigenvalues of B*_{t}*likewise must lie in these discs, it follows for reasons of continuity that also k eigenvalues*
*of A lie in M*_{1} *and the remaining n − k in M*_{2}.

*Remark 5.2.1 Take X = I, A = diag(A) + offdiag(A). Consider the transformation*
*A −→ 4*^{−1}*A4 with 4 = diag(δ*1*, · · · , δ**n**). The Gerschgorin discs:*

*D*_{i}*= {z ∈ C | |z − a*_{ii}*| ≤*
X*n*

*k=1*
*k6=i*

¯¯

¯¯*a*_{ik}*δ*_{k}*δ*_{i}

¯¯

¯*¯ =: ρ*^{i}*}.*

*Example 5.2.1 Let A =*

*1 ² ²*

*² 2 ²*

*² ² 2*

*, D*1 *= {z | |z − 1| ≤ 2²}, D*2 *= D*3 *= {z |*

*|z − 2| ≤ 2²}, 0 < ² ¿ 1. Transformation with 4 = diag(1, k², k²), k > 0 yields*
*A = 4*˜ ^{−1}*A4 =*

*1 k²*^{2} *k²*^{2}

1

*k* 2 *²*

1

*k* *²* 2

.

For ˜*A we have ρ*_{1} *= 2k²*^{2}*, ρ*_{2} *= ρ*_{3} = ^{1}_{k}*+ ². The discs D*_{1} *and D*_{2} *= D*_{3} for ˜*A are disjoint if*
*ρ*1*+ ρ*2 *= 2k²*^{2} +^{1}_{k}*+ ² < 1.*

*For this to be true we must clearly have k > 1. The optimal value ˜k, for which D*_{1} and
*D*_{2}(for ˜*A) touch one another, is obtained from ρ*_{1}*+ ρ*_{2} = 1. One finds

*˜k =* 2
*1 − ² +*p

*(1 − ²)*^{2}*− 8²*^{2} *= 1 + ² + O(²*^{2})

*and thus ρ*1 *= 2˜k²*^{2} *= 2²*^{2}*+ O(²*^{3}*). Through the transformation A −→ ˜A the radius ρ*1 of
*D*_{1} *can thus be reduced from the initial 2² to about 2²*^{2}.

*Theorem 5.2.2 (Bauer-Fike) If µ is an eigenvalue of A + E ∈ C*^{n×n}*and X*^{−1}*AX =*
*D = diag(λ*_{1}*, · · · , λ*_{n}*), then*

*λ∈σ(A)*min *|λ − µ| ≤ κ**p**(X) k E k**p**,*

*where k · k**p* *is p-norm and κ**p**(X) =k X k**p**k X*^{−1}*k**p* *.*

*Proof: We need only consider the case µ 6∈ σ(A). If X*^{−1}*(A + E − µI)X is singular, then*
*so is I + (D − µI)*^{−1}*(X*^{−1}*EX). Thus,*

*1 ≤k (D − µI)*^{−1}*(X*^{−1}*EX) k*_{p}*≤* 1

*λ∈σ(A)*min *|λ − µ|* *k X k*_{p}*k E k*_{p}*k X*^{−1}*k*_{p}*.*

*Theorem 5.2.3 Let Q*^{∗}*AQ = D+N be a Schur decomposition of A with D = diag(λ*_{1}*, · · · , λ** _{n}*)

*and N strictly upper triangular, N*

^{n}*= 0. If µ ∈ σ(A + E), then*

*λ∈σ(A)*min *|λ − µ| ≤ max{θ, θ*^{n}^{1}*},*
*where θ =k E k*_{2} P_{n−1}

*k=0* *k N k*^{k}_{2}*.*

*Proof: Define δ = min*_{λ∈σ(A)}*|λ − µ|. The theorem is true if δ = 0. If δ > 0, then*
*I − (µI − A)*^{−1}*E is singular and we have*

*1 ≤ k (µI − A)*^{−1}*E k*_{2}

*≤ k (µI − A)*^{−1}*k*2*k E k*2

*= k [(µI − D) − N]*^{−1}*k*_{2}*k E k*_{2} *.*

*Since (µI − D) is diagonal it follows that [(µI − D)*^{−1}*N]** ^{n}* = 0 and therefore

*[(µI − D) − N]** ^{−1}* =
X

*n−1*

*k=0*

*[(µI − D)*^{−1}*N]*^{k}*(µI − D)*^{−1}*.*

Hence we have

*1 ≤* *k E k*_{2}

*δ* *max{1,* 1
*δ*^{n−1}*}*

X*n−1*
*k=0*

*k N k*^{k}_{2}*,*

from which the theorem readily follows.

*Example 5.2.2 If A =*

1 2 3

0 4 5

*0 0 4.001*

* and E =*

0 0 0

0 0 0

*0.001 0 0*

*. Then σ(A + E) ∼*=
*{1.0001, 4.0582, 3.9427} and A’s matrix of eigenvectors satisfies κ*2*(X) ∼*= 10^{7}. The Bauer-
Fike bound in Theorem 5.2.2 has order 10^{4}, but the Schur bound in Theorem 5.2.3 has
order 10^{0}.

*Theorems 5.2.2 and 5.2.3 each indicate potential eigenvalue sensitively if A is non-*
*normal. Specifically, if κ*_{2}*(X) and k N k*^{n−1}_{2} *is large, then small changes in A can induce*
large change in the eigenvalues.

*Example 5.2.3 If A =*

· *0 I*9

0 0

¸

*and E =*

· 0 0

10* ^{−10}* 0

¸

*, then for all λ ∈ σ(A) and*
*µ ∈ σ(A + E), |λ − µ| = 10*^{−10}^{10} . So a change of order 10^{−10}*in A results in a change of*
order 10* ^{−1}* in its eigenvalues.

*Let λ be a simple eigenvalue of A ∈ C*^{n×n}*and x and y satisfy Ax = λx and y*^{∗}*A = λy*^{∗}*with k x k*2*=k y k*2= 1. Using classical results from Function Theory, it can be shown
*that there exists differentiable x(ε) and λ(ε) such that*

*(A + εF )x(ε) = λ(ε)x(ε)*

*with k x(ε) k*_{2}*= 1 and k F k*_{2}*≤ 1, and such that λ(0) = λ and x(0) = x. By differentiating*
*and set ε = 0:*

*A ˙x(0) + F x = ˙λ(0)x + λ ˙x(0).*

*Applying y*^{∗}*to both sides and dividing by y*^{∗}*x =⇒*

*f (x, y) = y*^{n}*+ p*_{n−1}*(x)y*^{n−1}*+ p*_{n−2}*(x)y*^{n−2}*+ · · · + p*_{1}*(x)y + p*_{0}*(x).*

*Fix x, then f (x, y) = 0 has n roots y*_{1}*(x), · · · , y*_{n}*(x). f (0, y) = 0 has n roots y*_{1}*(0), · · · , y** _{n}*(0).

*Theorem 5.2.4 Suppose y**i**(0) is a simple root of f (0, y) = 0, then there is δ**i* *> 0 such*
*that there is a simple root y*_{i}*(x) of f (x, y) = 0 defined by*

*y**i**(x) = y**i**(0) + p**i1**x + p**i2**x*^{2}*+ · · · ,* *(may terminate!)*
*where the series is convergent for |x| < δ**i**. (y**i**(x) −→ y**i**(0) as x −→ 0).*

*Theorem 5.2.5 If y*_{1}*(0) = · · · = y*_{m}*(0) is a root of multiplicity m of f (0, y) = 0, then*
*there exists δ > 0 such that there are exactly m zeros of f (x, y) = 0 when |x| < δ having*
*the following properties:*

(a) P_{r}

*i=1**m*_{i}*= m,* *m*_{i}*≥ 0. The m roots fall into r groups.*

*(b) Those roots in the group of m**i* *are m**i* *values of a series*
*y*_{1}*(0) + p*_{i1}*z + p*_{i2}*z*^{2} *+ · · ·*

*corresponding to the m*_{i}*different values of z defined by z = x*^{mi}^{1} *.*

*Let λ*_{1} *be a simple root of A and x*_{1} *be the corresponding eigenvector. Since λ*_{1} is
*simple, (A − λ*1*I) has at least one nonzero minor of order n − 1. Suppose this lies in the*
*first (n − 1) rows of (A − λ*_{1}*I). Take x*_{1} *= (A*_{n1}*, A*_{n2}*, · · · , A** _{nn}*). Then

*(A − λ*_{1}*I)*

*A*_{n1}*A** _{n2}*
...

*A*_{nn}

=

0 0...

0

*,*

since P_{n}

*j=1**a**nj**A**nj* *= det(A − λ*1*I) = 0. Here A**ni* *is the cofactor of a**ni*, hence it is a
*polynomial in λ*_{1} *of degree not greater than (n − 1).*

*Let λ*_{1}*(ε) be the simple eigenvalue of A + εF and x*_{1}*(ε) be the corresponding eigen-*
*vector. Then the elements of x*1*(ε) are the polynomial in λ*1*(ε) and ε. Since the power*
*series for λ*_{1}*(ε) is convergent for small ε, so x*_{1}*(ε) = x*_{1}*+ εz*_{1}*+ ε*^{2}*z*_{2}*+ · · · is a convergent*
power series

¯¯

*¯ ˙λ(0)*

¯¯

¯ = *|y*^{∗}*F x|*

*|y*^{∗}*x|* *≤* 1

*|y*^{∗}*x|. The upper bound is attained if F = yx** ^{∗}*. We refer

*to the reciprocal of s(λ) ≡ |y*

^{∗}*x| as the condition number of the eigenvalue λ.*

*λ(ε) = λ(0) + ˙λ(0)ε + O(ε*^{2}*), an eigenvalue λ may be perturbed by an amount* *ε*
*s(λ)*,
*if s(λ) is small then λ is appropriately regarded as ill-conditioned. Note that s(λ) is*
*the cosine of the angle between the left and right eigenvectors associated with λ and is*
*unique only if λ is simple. A small s(λ) implies that A is near a matrix having a multiple*
*eigenvalue. In particular, if λ is distinct and s(λ) < 1, then there exists an E such that*
*λ is a repeated eigenvalue of A + E and*

*k E k*_{2}*≤* *s(λ)*
p*1 − s*^{2}*(λ),*
this is proved in Wilkinson(1972).

*Example 5.2.4 If A =*

1 2 3

0 4 5

*0 0 4.001*

* and E =*

0 0 0

0 0 0

*0.001 0 0*

*. Then σ(A + E) ∼*=
*{1.0001, 4.0582, 3.9427} and s(1) ∼= 0.79 × 10*^{0}*, s(4) = 0.16 × 10*^{−3}*, s(4.001) ∼= 0.16 × 10** ^{−3}*.

*Observe that k E k*2

*/s(λ) is a good estimate of the perturbation that each eigenvalue*undergoes.

*If λ is a repeated eigenvalue, then the eigenvalue sensitivity question is more compli-*
*cated. For example A =*

· *1 a*
0 1

¸

*and F =*

· 0 0 1 0

¸

*then σ(A + εF ) = {1 ±√*

*εa}. Note*
*that if a 6= 0 then the eigenvalues of A + εF are not differentiable at zero, their rate of*
*change at the origin is infinite. In general, if λ is a detective eigenvalue of A, then O(ε)*
*perturbations in A result in O(ε*^{p}^{1}*) perturbations in λ where p ≥ 2 (see Wilkinson AEP*
pp.77 for a more detailed discussion).

*We now consider the perturbations of invariant subspaces. Assume A ∈ C** ^{n×n}* has

*distinct eigenvalues λ*

_{1}

*, · · · , λ*

_{n}*and k F k*

_{2}= 1. We have

*(A + εF )x*_{k}*(ε) = λ*_{k}*(ε)x*_{k}*(ε),* *k x*_{k}*(ε) k*_{2}*= 1,*

and

*y*_{k}^{∗}*(ε)(A + εF ) = λ**k**(ε)y*^{∗}_{k}*(ε),* *k y**k**(ε) k*2*= 1,*

*for k = 1, · · · , n, where each λ**k**(ε),x**k**(ε) and y**k**(ε) are differentiable. Set ε = 0 :*
*A ˙x**k**(0) + F x**k* *= ˙λ**k**(0)x**k**+ λ**k**˙x**k**(0),*

*where λ**k* *= λ**k**(0) and x**k* *= x**k**(0). Since {x**i**}*^{n}_{i=1}*linearly independent, write ˙x**k*(0) =
P_{n}

*i=1**a*_{i}*x** _{i}*, so we have
X

*n*

*i=1*
*i6=k*

*a**i**(λ**i**− λ**k**)x**i**+ F x**k**= ˙λ**k**(0)x**k**.*

*But y*_{i}^{∗}*(0)x*_{k}*= y*_{i}^{∗}*x*_{k}*= 0, for i 6= k and thus*

*a*_{i}*= y*_{i}^{∗}*F x*_{k}*/[(λ*_{k}*− λ*_{i}*)y*_{i}^{∗}*x*_{i}*],* *i 6= k.*

*Hence the Taylor expansion for x*_{k}*(ε) is*

*x*_{k}*(ε) = x*_{k}*+ ε*
X*n*

*i=1*
*i6=k*

½ *y*_{i}^{∗}*F x*_{k}*(λ*_{k}*− λ*_{i}*)y*_{i}^{∗}*x*_{i}

¾

*x*_{i}*+ O(ε*^{2}*).*

*Thus the sensitivity of x**k* *depends upon eigenvalue sensitivity and the separation of λ**k*

from the other eigenvalues.

*Example 5.2.5 If A =*

· *1.01 0.01*
*0.00 0.99*

¸

*, then λ = 0.99 has Condition* 1

*s(0.99)* *∼= 1.118*
*and associated eigenvector x = (0.4472, −8.944)** ^{T}*. On the other hand, ˜

*λ = 1.00 of the*

*”nearby” matrix A + E =*

· *1.01 0.01*
*0.00 1.00*

¸

has an eigenvector ˜*x = (0.7071, −0.7071)*^{T}*.*
Suppose

*Q*^{∗}*AQ =*

· *T*_{11} *T*_{12}
0 *T*_{22}

¸ *}p*

*}q = n − p* (5.2.1)

*is a Schur decomposition of A with*

*Q = [ Q*|{z}_{1}

*p*

*, Q*|{z}_{2}

*n−p*

*].* (5.2.2)

*Definition 5.2.1 We define the separation between T*11 *and T*22 *by*
*sep*_{F}*(T*_{11}*, T*_{22}) = min

*Z6=0*

*k T*_{11}*Z − ZT*_{22} *k*_{F}*k Z k*_{F}*.*

*Definition 5.2.2 Let X be a subspace of C*^{n}*, X is called an invariant subspace of A ∈*
C^{n×n}*, if AX ⊂ X (i.e. x ∈ X =⇒ Ax ∈ X).*

*Theorem 5.2.6 A ∈ C*^{n×n}*, V ∈ C*^{n×r}*and rank(V ) = r, then there are equivalent:*

*(a) there exists S ∈ C*^{r×r}*such that AV = V S.*

*(b) R(V ) is an invariant subspace of A.*

Proof: Trivial!

*Remark 5.2.2 (a) If Sz = µz, z 6= 0 then µ is eigenvalue of A with eigenvector V z.*

*(b) If V is a basis of X, then ˜V = V (V*^{∗}*V )*^{−}^{1}^{2} *is an orthogonal basis of X.*

*Theorem 5.2.7 A ∈ C*^{n×n}*, Q = (Q*_{1}*, Q*_{2}*) orthogonal, then there are equivalent:*

*(a) If Q*^{∗}*AQ = B =*

· *B*_{11} *B*_{12}
*B*21 *B*22

¸

*, then B*_{21}*= 0.*

*(b) R(Q*_{1}*) is an invariant subspace of A.*

*Proof: Q*^{∗}*AQ = B ⇐⇒ AQ = QB = (Q*_{1}*, Q*_{2})

· *B*_{11} *B*_{12}
*B*21 *B*22

¸

*. Thus AQ*_{1} *= Q*_{1}*B*_{11}+
*Q*_{2}*B*_{21}.

*(a) B*_{21}*= 0, then AQ*_{1} *= Q*_{1}*B*_{11}*, so R(Q*_{1}*) is an invariant subspace of A (from Theorem*
5.2.6).

*(b) R(Q*1*) is invariant subspace. There exists S such that AQ*1 *= Q*1*S = Q*1*B*11*+Q*2*B*21.
*Multiply with Q*^{∗}_{1}, then

*S = Q*^{∗}_{1}*Q*_{1}*S = Q*^{∗}_{1}*Q*_{1}*B*_{11}*+ Q*^{∗}_{1}*Q*_{2}*B*_{21}*.*
*So S = B*11 *=⇒ Q*2*B*21 *= 0 =⇒ Q*^{∗}_{2}*Q*2*B*21 *= 0 =⇒ B*21= 0.

*Theorem 5.2.8 Suppose (5.2.1) and (5.2.2) hold and for E ∈ C*^{n×n}*we partition Q*^{∗}*EQ*
*as follows:*

*Q*^{∗}*EQ =*

· *E*11 *E*12

*E*_{21} *E*_{22}

¸

*with E*_{11} *∈ R*^{p×p}*and E*_{22} *∈ R**(n−p)×(n−p)**. If*

*δ = sep*2*(T*11*, T*22*)− k E*11 *k*2 *− k E*22*k*2*> 0*
*and*

*k E*_{21}*k*_{2} *(k T*_{12}*k*_{2} *+ k E*_{12} *k*_{2}*) ≤ δ*^{2}*/4.*

*Then there exists P ∈ C*^{(n−k)×k}*such that*

*k P k*_{2}*≤ 2 k E*_{21} *k*_{2} */δ*

*and such that the column of ˜Q*_{1} *= (Q*_{1}*+ Q*_{2}*P )(I + P*^{∗}*P )*^{−}^{1}^{2} *form an orthonormal basis*
*for a invariant subspace of A + E.(See Stewart 1973).*

*Lemma 5.2.1 Let {s*_{m}*} and {p*_{m}*} be two sequence defined by*

*s*_{m+1}*= s*_{m}*/(1 − 2ηp*_{m}*s*_{m}*),* *p*_{m+1}*= ηp*^{2}_{m}*s*_{m+1}*,* *m = 0, 1, 2, · · ·* (5.2.3)
*and*

*s*0 *= σ,* *p*0 *= σγ* (5.2.4)

*satisfying*

*4ησ*^{2}*γ < 1. (Here σ, η, γ > 0)* (5.2.5)
*Then {s*_{m}*} is monotonic increasing and bounded above; {p*_{m}*} is monotonic decreasing,*
*converges quadratically to zero.*

Proof: Let

*x**m* *= s**m**p**m**,* *m = 0, 1, 2, · · · .* (5.2.6)
From (5.2.3) we have

*x*_{m+1}*= s*_{m+1}*p*_{m+1}*= ηp*^{2}_{m}*s*^{2}_{m}*/(1 − 2ηp*_{m}*s** _{m}*)

^{2}

*= ηx*

^{2}

_{m}*/(1 − 2ηx*

*)*

_{m}^{2}

*,*(5.2.7) (5.2.5) can be written as

*0 < x*0 *<* 1

*4η. (since x*0 *= s*0*p*0 *= σ*^{2}*γ <* 1

*4η*) (5.2.8)

Consider

*x = f (x),* *f (x) = ηx*^{2}*/(1 − 2ηx)*^{2}*,* *x ≥ 0.* (5.2.9)
By

*df (x)*

*dx* = *2ηx*

*(1 − 2ηx)*^{3}*,*

*we know that f (x) is differentiable and monotonic increasing in [0, 1/2η), and* *df (x)*

*dx* *|** _{x=0}*= 0

*: The equation (5.2.9) has zeros 0 and 1/4η in [0, 1/2η). Under Condition (5.2.8) the*

*iteration x*

*m*as in (5.2.7) must be monotone decreasing converges quadratically to zero.

*(Issacson &Keller ”Analysis of Num. Method 1996, Chapter 3 §1.) Thus*
*s*_{m+1}

*s** _{m}* = 1

*1 − 2ηx** _{m}* = 1 +

*2ηx*

_{m}*1 − 2ηx*_{m}*= 1 + t**m**,*

*where t** _{m}* is monotone decreasing, converges quadratically to zero, hence

*s*_{m+1}*= s*_{0}
Y*m*
*j=0*

*s**j+1*

*s*_{j}*= s*_{0}
Y*m*
*j=0*

*(1 + t** _{j}*)

*monotone increasing, and converges to s*_{0}
Y*∞*
*j=0*

*(1 + t*_{j}*) < ∞, so p** _{m}* =

*x*

_{m}*s** _{m}* monotone de-
creasing, and quadratically convergent to zero.

*Theorem 5.2.9 Let*

*P A*_{12}*P + P A*_{11}*− A*_{22}*P − A*_{21} = 0 (5.2.10)
*be the quadratic matrix equation in P ∈ C*^{(n−l)×l}*(1 ≤ l ≤ n), where*

· *A*_{11} *A*_{12}
*A*_{21} *A*_{22}

¸

*= A,* *σ(A*_{11})\

*σ(A*_{22}*) = ∅.*

*Define operator T by:*

*T Q ≡ QA*_{11}*− A*_{22}*Q,* *Q ∈ C*^{(n−l)×l}*.* (5.2.11)
*Let*

*η =k A*12*k,* *γ =k A*21*k* (5.2.12)

*and*

*σ =k T*^{−1}*k= sup*

*kP k=1*

*k T*^{−1}*P k .* (5.2.13)

*If*

*4ησ*^{2}*γ < 1,* (5.2.14)

*then according to the following iteration, we can get a solution P of (5.2.10) satisfying*

*k P k≤ 2σγ,* (5.2.15)

*and this iteration is quadratic convergence.*

*Iteration: Let A**m* =

"

*A*^{(m)}_{11} *A*^{(m)}_{12}
*A*^{(m)}_{21} *A*^{(m)}_{22}

#

*,* *A*0 *= A.*

*(i) Solve*

*T*_{m}*P*_{m}*≡ P*_{m}*A*^{(m)}_{11} *− A*^{(m)}_{22} *P*_{m}*= A*^{(m)}_{21} (5.2.16)
*and get* *P*_{m}*∈ C*^{(n−l)×l}*;*

*(ii) Compute*

*A*^{(m+1)}_{11} *= A*^{(m)}_{11} *+ A*_{12}*P*_{m}*,*
*A*^{(m+1)}_{22} *= A*^{(m)}_{22} *− P**m**A*12*,*
*A*^{(m+1)}_{21} *= −P*_{m}*A*_{12}*P*_{m}*.*
*Goto (i), solve P*_{m+1}*.*

*Then*

*P = lim*

*m→∞*

X*m*
*i=0*

*P**i* (5.2.17)

*is a solution of (5.2.10) and satisfies (5.2.15).*

*Proof: (a) Prove that for m = 0, 1, 2, · · · ,* *T*_{m}* ^{−1}* exist: denote

*k T*_{m}^{−1}*k= σ*_{m}*,* *(T = T*_{0}*,* *σ = σ*_{0}*),* (5.2.18)
then

*4 k A*12 *kk P**m* *k σ**m* *< 1.* (5.2.19)
*By induction, m = 0, from σ(A*_{11})T

*σ(A*_{22}*) = ∅ we have T*_{0} *= T is nonsingular. From*
(5.2.12)-(5.2.14) it holds

*4 k A*_{12} *kk P*_{0} *k σ*_{0} *= 4η k T*^{−1}*A*_{21} *k σ ≤ 4ησ*^{2}*γ < 1.*

*Suppose T*_{m}^{−1}*exists, and (5.2.19) holds, prove that T*_{m+1}* ^{−1}* exists and

*4 k A*12

*kk P*

*m+1*

*k σ*

*m+1*

*< 1.*

From the definition

*sep(A*11*, A*22) = inf

*kQk=1**k QA*11*− A*22*Q k*

*and the existence of T*^{−1}*follows sep(A*_{11}*, A*_{22}*) =k T*^{−1}*k*^{−1}*= σ** ^{−1}*, and by the perturbation
property of ”sep” follows

*sep(A*^{(m+1)}_{11} *, A*^{(m+1)}_{22} *) = sep(A*^{(m)}_{11} *+ A*12*P**m**, A*^{(m)}_{22} *− P**m**A*12)

*≥ sep(A*^{(m)}_{11} *, A*^{(m)}_{22} *)− k A*_{12}*P*_{m}*k − k P*_{m}*A*_{12}*k*

*≥* *1 − 2 k A*_{12}*kk P*_{m}*k σ*_{m}*σ**m*

*> 0.* (5.2.20)

From

*sep(A*_{11}*, A*_{22}*) ≤ min{|λ*_{1}*− λ*_{2}*| : λ*_{1} *∈ σ(A*_{11}*), λ*_{2} *∈ σ(A*_{22}*)}.*

*We have σ(A*^{(m+1)}_{11} )T

*σ(A*^{(m+1)}_{22} *) = ∅, hence T*_{m+1}* ^{−1}* exists and

*sep(A*

^{(m+1)}_{11}

*, A*

^{(m+1)}_{22}

*) =k T*

_{m+1}

^{−1}*k*

^{−1}*= σ*

^{−1}

_{m+1}*.*From (5.2.20) it follows

*σ*_{m+1}*≤* *σ*_{m}

*1 − 2 k A*_{12} *kk P*_{m}*k σ*_{m}*.* (5.2.21)
*Substitute (5.2.19) into (5.2.21), we get σ*_{m+1}*≤ 2σ** _{m}*, and

*k P*_{m+1}*k≤k T*_{m+1}^{−1}*kk A*^{m+1}_{21} *k≤ σ*_{m+1}*k P*_{m}*k*^{2}*k A*_{12}*k<* 1

2 *k P*_{m}*k .*
Hence

*2 k A*_{12} *kk P*_{m+1}*k σ*_{m+1}*≤ 2 k A*_{12} *kk P*_{m}*k σ*_{m}*< 1/2.*

*This proved that T*_{m}^{−1}*exists for all m = 0, 1, 2, · · · and (5.2.19) holds.*

*(b) Prove k P*_{m}*k is quadratic convergence to zero. Construct sequences {q*_{m}*}, {s*_{m}*}, {p*_{m}*}*
satisfying

*k A*^{(m)}_{21} *k≤ q**m**,* *σ**m* *≤ s**m**,* *k P**m* *k≤ p**m**.* (5.2.22)
From

*A*^{(m+1)}_{21} *= −P**m**A*12*P**m* (5.2.23)

follows

*k A*^{(m+1)}_{21} *k≤k A*12*kk P**m* *k*^{2}*≤ ηp*^{2}_{m}*.* (5.2.24)
*Define {q**m**} by*

*q**m+1* *= ηp*^{2}_{m}*,* *q*0 *= γ;* (5.2.25)
From (5.2.21) we have

*σ*_{m+1}*≤* *s*_{m}

*1 − 2ηp*_{m}*s*_{m}*.* (5.2.26)

*Define {s*_{m}*} by*

*s** _{m+1}* =

*s*

_{m}*1 − 2ηp*_{m}*s*_{m}*,* *s*_{0} *= σ;* (5.2.27)
From (5.2.16) we have

*k P**m* *k≤k T*_{m}^{−1}*kk A*^{(m)}_{21} *k= σ**m* *k A*^{(m)}_{21} *k≤ s**m**q**m**.*
*Define {p*_{m}*} by*

*p*_{m+1}*= s*_{m+1}*q*_{m+1}*= ηp*^{2}_{m}*s*_{m+1}*,* *p*_{0} *= σγ.* (5.2.28)
*By Lemma 5.2.1 follows that {p*_{m}*} & 0 monotone and form (5.2.22) follows that k*
*P**m* *k−→ 0 quadratically.*

*(c) Prove P*^{(m)}*−→ P and (5.2.15) holds. According to the method as in Lemma*
*5.2.1. Construct {x*_{m}*} (see (5.2.6),(5.2.7) ), that is*

*x** _{m+1}* =

*ηx*

^{2}

_{m}*(1 − 2ηx** _{m}*)

^{2}

*,*

*s*

*=*

_{m+1}*s*

*m*

*1 − 2ηx** _{m}* (5.2.29)

and then

*p** _{m+1}* =

*x*

_{m+1}*s** _{m+1}* =

*ηx*

_{m}*1 − 2ηx*_{m}*p*_{m}*.* (5.2.30)

*By induction! For all m = 1, 2, · · · we have*
*p*_{m}*<* 1

2*p*_{m−1}*,* *x*_{m}*<* 1

*4η.* (5.2.31)

In fact, substitute

*ηx*_{0}

*1 − 2ηx*_{0} = *ησ*^{2}*γ*

*1 − 2ησ*^{2}*γ* *<* 1

2 (5.2.32)

*into (5.2.30) and get p*1 *<* 1

2*p*0; From (5.2.29) and (5.2.32) it follows that
*x*_{1} = 1

*η*

µ *ηx*0

*1 − 2ηx*_{0}

¶_{2}

*<* 1
*4η.*