Hence we have|λ(L+ ∆L)−λ(L)| ≤cond(λ,L)|||∆L|||w,p+O((|||∆L|||w,p)2).
(b) LetH∈Lpw(Cn×n,k · k) be such that|||H|||w,p = 1.Then the partial condition operator DH : M(Lpw(Cn×n,k · k),C) → C,(λ,L) 7→ DHλ(L) and the partial condition number condH(λ,L) are given by
DHλ(L) = y∗H(λ)x
y∗Bx =hH, K(λ,L)iL and condH(λ,L) = |y∗H(λ)x|
|y∗Bx| =|hH, K(λ,L)iL|.
Hence for small |t|, we have |λ(L+tH)−λ(L)| ≤condH(λ,L)|t|+O(|t|2).
(c) For a subordinate normk · k on Cn×n,choose u, v ∈Cn such that kuk= 1, kvk∗ = 1, y∗u=kyk∗ and v∗x=kxk. Now define
H1 :=−∇1(k(1, λ)kw−1,q)uv∗ and H2 :=∇2(k(1, λ)kw−1,q)uv∗
and consider the pencil H(z) :=H1−zH2, where p−1+q−1 = 1. Here ∇i(k(1, λ)kw−1,q) is the partial gradient of the map C2 → R,(z1, z2) 7→ k(z1, z2)kw−1,q evaluated at (1, λ).
Then we have |||H|||w,p = 1 and
condH(λ,L) =|hH, K(λ,L)iL|=kK(λ,L)kw−1,q = cond(λ,L).
Proof: We only need to prove (c). First note that kH1k = |∇1(k(1, λ)kw−1,q)| and kH2k=|∇2(k(1, λ)kw−1,q)|.Hence |||H|||w,p =k(∇1(k(1, λ)kw−1,q), ∇2(k(1, λ)kw−1,q))kw,p. By Lemma 2.4.3, we havek(∇1(k(1, λ)kw−1,q), ∇2(k(1, λ)kw−1,q))kw,p = 1.Hence|||H|||w,p = 1. Now condH(λ,L) = |hH, K(λ,L)iL|=|y∗H(λ)x
y∗Bx |. Again by Lemma 2.4.3, we have
|y∗H(λ)x|= (|∇1(k(1, λ)kw−1,q) +λ∇2(k(1, λ)kw−1,q)|)kyk∗kxk=k(1, λ)kw−1,qkyk∗kxk.
Hence the result follows. ¥
Remark 5.5.5. For the case when one of the components of w is 0, the results in Theorem 5.5.4 still hold. However, in such a case, these results are valid only for the w-admissible perturbation ofL.
5.6 Sensitivity of simple eigenvalues of matrix poly-
is an inner product on Lm(Cn×n), where L1(z) = Pm
i=0ziAi and L2(z) = Pm
i=0ziBi. Consequently, if F : Lm(Cn×n) → C is a linear functional then there is a unique G ∈ Lm(Cn×n) such that F(X) = hX, GiL for allX ∈L(Cn×n).Again recall that the action ofRm+1 onLm(Cn×n) is the mapRm+1×Lm(Cn×n)→Lm(Cn×n),(w,L)7→w¯Lgiven by (w¯L)(z) =Pm
i=0ziwiAi, whereL(z) = Pm
i=0ziAi.Now consider the normed space of polynomialsLpm,w(Cn×n,k · k).Then we have the following.
Theorem 5.6.1. Consider the space Lpm,w(Cn×n,k · k). Suppose that each component of w is nonzero. Then (Lpm,w(Cn×n,k · k))∗ = (Lqm,w−1(Cn×n,k · k∗)), where p−1+q−1 = 1 and k · k∗ is the dual norm of the norm k · k on Cn×n.
Proof: The proof is similar to that of Theorem 5.5.2. ¥ We now define condition polynomial.
Definition 5.6.2. Let f ∈CL1(Lpm,w(Cn×n,k · k),C) andDf(L) be the derivative of f at L. If there is a matrix polynomial K(f,L) ∈ Lpm,w−1(Cn×n,k · k∗) such that Df(L)Y = hY, K(f,L)iL for all Y ∈ Lpm,w(Cn×n,k · k) then K(f,L) is said to be the condition polynomial of (f,L).
Let L ∈ Lpm,w(Cn×n,k · k) be regular. Let (λ, y, x) be a simple eigentriple of L. As before we writeλasλ(L) and treatλas a function ofL.Now following similar arguments as those given for matrix pencils, it follows that there is an open setU ⊂Lpm,w(Cn×n,k·k) containingLsuch that the mapλ:U →C, L7→λ(L) defines a smooth function. Indeed, we have the following result.
Theorem 5.6.3. LetL∈Lpm,w(Cn×n,k·k)be regular. Let(λ, y, x)be a simple eigentriple ofL.For H∈Lpm,w(Cn×n,k·k),consider the one parameter family of matrix polynomials L+tH, t∈C. Then there is a finite set F ⊂C and an eigenvalueλ(L+tH) of L+tH such that λ(L+tH) is simple and holomorphic for all t∈C\F. Further, we have
λ(L+tH) = λ(L)− y∗H(λ)x
y∗∂λL(λ)xt+O(|t|2) for small |t|. Furthermore, we have λ∈CL1¡
Lpm,w(Cn×n,k · k), C¢ and λ(L+H) =λ(L)− y∗H(λ)x
y∗∂λL(λ)x +O((|||H|||w,p)2)
for sufficiently small |||H|||w,p. Here∂λL(λ) is the derivative of the polynomial L(z)eval- uated at λ.
Proof: The proof is similar to that of Theorem 5.5.1. Indeed, set λ(t) := λ(L + tH) = λ + αt + O(t2) and x(t) = x + x1t + O(t2) be an associated right eigen- vector. Now L(λ(t)) = L(λ) +∂λL(λ)αt +O(t2). Hence y∗L(λ(t)) = y∗∂λL(λ)αt+
O(t2) ⇒ y∗L(λ(t))x(t) = y∗∂λL(λ)xαt+ O(t2). Similarly, H(λ(t)) = H(λ) + O(t).
Hence y∗H(λ(t))x(t) = y∗H(λ)x + O(t). Now y∗(L(λ(t))x(t) +tH(λ(t))x(t) = 0 ⇒ αty∗∂λL(λ)x+ty∗H(λ)x+O(t2) = 0. This shows that α=− y∗H(λ)x
y∗∂λL(λ)x +O(t). Con- sequently, we haveλ(t) = λ− y∗H(λ)x
y∗∂λL(λ)x+O(t2). Hence the result follows. ¥ We mention that the first order expansion of λ(t) is well known [46, 18].
Now, let L ∈ Lpm,w(Cn×n,k · k) be a regular polynomial and (λ, y, x) be a simple eigentriple of L. Then λ ∈ CL1(Lpm,w(Cn×n,k · k), C) and there is a unique condition polynomial K(λ,L) ∈ Lqm,w−1(Cn×n,k · k∗) such that Dλ(L) = hX, K(λ,L)iL for all X ∈Lpm,w(Cn×n,k·k).Hence cond(λ,L) = kDλ(L)k=|||K(λ,L)|||w−1,q.DefineK(λ,L)∈ Lqm,w−1(Cn×n,k · k∗) by
K(λ,L)(z) :=
Xm
i=0
ziKi, whereKi :=− λiyx∗ y∗∂λL(λ)x. Then by Theorem 5.6.3, we have
Dλ(L)H=− y∗H(λ)x
y∗∂λL(λ)x =hH, K(λ,L)iL
for all H∈Lpm,w(Cn×n,k · k). Indeed, suppose that H(z) = Pm
i=0ziHi. Then
− y∗H(λ)x
y∗∂λL(λ)x =−(hH0, yx∗/αi+hH1, λ yx∗/αi · · ·+hHm, λmyx∗/αi) = hH, K(λ,L)iL, where α := y∗∂λL(λ)x. In particular, for H ∈ Lpm,w(Cn×n,k · k) with |||H|||w,p = 1, we have DHλ(L) =hH, K(λ,L)iL =−yy∗∗∂H(λ)xλL(λ)x.
Thus, we have
cond(λ,L) =|||K(λ,L)|||w−1,q = k(1, λ, . . . , λm)kw−1,qkyx∗k∗
|y∗∂λL(λ)x| ,
wherep−1+q−1 = 1.When the norm k · kon Cn×n is a subordinate norm, we have cond(λ,L) = k(1, λ, . . . , λm)kw−1,qkyk∗kxk
|y∗∂λL(λ)x| .
Theorem 5.6.4. Let L∈Lpm,w(Cn×n,k · k) be regular and p−1+q−1 = 1. Suppose that each component of w is nonzero. Let (λ, y, x) be a simple eigentriple of L.
(a) The condition operatorD :M¡
Lpm,w(Cn×n,k · k), C¢
→(Lpm,w(Cn×n,k·k))∗,(λ,L)7→
Dλ(L)is given byDλ(L)X :=hX, K(λ,L)iL forX ∈Lpm,w(Cn×n,k·k),whereK(λ,L)∈ Lqm,w−1(Cn×n,k · k∗) is the condition polynomial given by
K(λ,L)(z) :=
Xm
i=0
ziKi and Ki :=− λiyx∗ y∗∂λL(λ)x.
The condition number cond(λ,L) is given by
cond(λ,L) =|||K(λ,L)|||w−1,q = k(1, λ, . . . , λm)kw−1,qkyx∗k∗
|y∗∂λL(λ)x| . When the norm k · k on Cn×n is a subordinate norm,
cond(λ,L) = k(1, λ, . . . , λm)kw−1,qkyk∗kxk
|y∗∂λL(λ)x| .
Hence we have|λ(L+ ∆L)−λ(L)| ≤cond(λ,L)|||∆L|||w,p+O((|||∆L|||w,p)2).
(b) Let H ∈ Lpm,w(Cn×n,k · k) be such that |||H|||w,p = 1. Then the partial condition operator DH :M(Lpm,w(Cn×n,k · k), C)→C,(λ,L)7→DHλ(L) and the partial condition number condH(λ,L) are given by
DHλ(L) = − y∗H(λ)x
y∗∂λL(λ)x =hH, K(λ,L)iL and condH(λ,L) = |y∗H(λ)x|
|y∗∂λL(λ)x|. Hence for small |t|, we have |λ(L+tH)−λ(L)| ≤condH(λ,L)|t|+O(|t|2).
(c) For a subordinate normk · k on Cn×n,choose u, v ∈Cn such that kuk= 1, kvk∗ = 1, y∗u=kyk∗ and v∗x=kxk. Now define
Hi :=−∇i(k(1, λ, . . . , λm)kw−1,q)uv∗, i= 0,1, . . . , m, and consider the polynomial H(z) := Pm
i=0ziHi. Here ∇i(k(1, λ, . . . , λm)kw−1,q) is the partial gradient of the map Cm+1 → R,(z0, z1, . . . , zm) 7→ k(z0, z1, . . . , zm)kw−1,q evalu- ated at (1, λ, . . . , λm). Then we have |||H|||w,p = 1 and
condH(λ,L) =|hH, K(λ,L)iL|=kK(λ,L)kw−1,q = cond(λ,L).
Proof: We only need to prove (c). Note that kHik = |∇i(k(1, λ, . . . , λm)kw−1,q)| and
|||H|||w,p =k(∇0(k(1, λ, . . . , λm)kw−1,q), . . . ,∇m(k(1, λ, . . . , λm)kw−1,q))kw,p.By Lemma 3.5.4, we havek(∇0(k(1, λ, . . . , λm)kw−1,q), . . . ,∇m(k(1, λ, . . . , λm)kw−1,q))kw,p = 1.Hence|||H|||w,p = 1. Now condH(λ,L) = |hH, K(λ,L)iL|=| y∗H(λ)x
y∗∂λL(λ)x|. By Lemma 3.5.4, we have
|y∗H(λ)x| = Ã
| Xm
i=0
λi∇i(k(1, λ, . . . , λm)kw−1,q)|
!
kyk∗kxk
= k(1, λ, . . . , λm)kw−1,qkyk∗kxk.
Hence the result follows. ¥
For most practical purposes one is only interested in knowing cond(λ,L). The crux of the matter is that when L ∈Lpm,w(Cn×n,k · k) and λ is a simple eigenvalue of L, the sensitivity ofλ is measured by
cond(λ,L) = sup
|||∆L|||w,p=1
|lim
t→0
λ(L+t∆L)−λ(L)
t |=|||K(λ,L)|||w−1,q.
The sensitivity analysis of eigenvalues of matrix pencils for the case when some components of ware zero (which corresponds to some coefficients of the matrix polyno- mials remaining unperturbed) follows from Theorem 5.6.4 when the perturbations are restricted to bew-admissible.
We mention the similar results holds for homogeneous pencils and homogeneous ma- trix polynomials. Indeed, letL∈Lpm,w(Cn×n,k·k) be regular and ((λ, µ), y, x) be a simple eigentriple ofL.Now forH∈Lpm,w(Cn×n,k·k),let (λ(t), µ(t)) := (λ(L+tH), µ(L+tH)) be the eigenvalue of L+tH such that (λ(t), µ(t)) is the best approximation of (λ, µ), that is, (λ(t)−λ, µ(t)−µ) is orthogonal to (λ, µ). Then it is easily seen that the maps t 7→ (λ(t), µ(t)) and H 7→ (λ(L+H), µ(L+H)) are smooth functions. Now writing λ(t) =λ+α1t+O(t2), µ(t) =µ+α2t+O(t2) and x(t) := x+x1t+O(t2),where x(t) is a right eigenvector of L+tH corresponding to (λ(t), µ(t)), and arguing on the same lines as in Theorem 5.6.3, a little calculation shows that
α1 = − µy∗H(λ, µ)x
y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x +O(t), α2 = − λy∗H(λ, µ)x
y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x +O(t).
Now defineK(λ, µ,L)∈Lqm,w−1(Cn×n,k · k∗) by K(λ, µ,L)(c, s) :=
Xm
i=0
cm−isiKi, where Ki :=− λm−iµi yx∗
(y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x)∗. Then it follows that for H∈Lpm,w(Cn×n,k · k),we have
Dλ(L)H = − µy∗H(λ, µ)x
y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x =hH, µ K(λ, µ,L)iL, Dµ(L)H = − λy∗H(λ, µ)x
y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x =hH, λ K(λ, µ,L)iL.
Consequently, the individual sensitivity ofλandµare measured by|µ| |||K(λ, µ,L)|||w−1,q and |λ| |||K(λ, µ,L)|||w−1,q, respectively. This shows that
cond(λ, µ,L) := sup
|||∆L|||w,p=1
limt→0
k(λ(L+t∆L), µ(L+t∆L))−(λ(L), µ(L))k2 tk(λ, µ)k2
= |||K(λ, µ,L)|||w−1,q = k(λm, λm−1µ, . . . , µm)kw−1,qkyx∗k∗
|y∗(¯µ∂cL(λ, µ)−λ∂¯ sL(λ, µ))x| .
Finally, for a subordinate norm k · k on Cn×n, choose u, v ∈ Cn such that kuk = 1, kvk∗ = 1, y∗u=kyk∗ and v∗x=kxk. Now define
Hi :=−∇i(k(λm, λm−1µ, . . . , µm)kw−1,q)uv∗, i= 0,1, . . . , m
and considerH(c, s) :=Pm
i=0cm−isiHi,where∇i(k(λm, λm−1µ, . . . , µm)kw−1,q) is the par- tial gradient of the map Cm+1 → R,(z0, z1, . . . , zm) 7→ k(z0, z1, . . . , zm)kw−1,q evaluated at (λm, λm−1µ, . . . , µm).Then we have |||H|||w,p = 1 and
condH(λ, µ,L) := lim
t→0
k(λ(L+tH), µ(L+tH))−(λ(L), µ(L))k2
tk(λ(L), µ(L))k2
= | y∗H(λ, µ)x
y∗(µ∂cL(λ, µ))−λ∂sL(λ, µ))x|
= |hH, K(λ, µ,L)iL|=|||K(λ, µ,L)|||w−1,q
= k(λm, λm−1µ, . . . , µm)kw−1,qkyx∗k∗
|y∗(¯µ∂cL(λ, µ)−λ∂¯ sL(λ, µ))x|
= cond(λ, µ,L).
Note that various condition numbers discussed in the introduction of this chapter follow as special cases by choosing appropriate norms and weights.
Conclusion
We have developed a general framework for defining and analyzing pseudospectra of matrix pencils and matrix polynomial that unifies various definitions of pseudospectra of matrix pencils and polynomials proposed in the literature. We have shown that pseudospectra of matrix pencils/polynomials can be analyzed on the same lines as those of matrices.
We have undertaken a detailed analysis of backward error functions associated with matrix pencils and matrix polynomials. We have introduced a notion of critical points of backward errors of approximate eigenvalues of matrix pencils and shown that each critical point is a multiple eigenvalue of an appropriately perturbed pencil. We have shown that a minimal critical point can be read off from the pseudospectra of matrix pencils/polynomials. Further, we have shown that a solution of Wilkinson’s problem for matrix pencils/polynomials can be read off from the pseudospectra of matrix pen- cils/polynomials. Further, for a diagonal pencil we have provided a simple procedure for the construction of nearest defective pencils.
We have derived various pseudospectra inclusions for matrix pencils and have shown that pseudospectra inclusions provides a powerful geometric framework for analyzing stability of eigendecompositions. We have introduced analogues of various separation of matrices to the case of matrix pencils and have shown their power and usefulness in analyzing stability of eigendecompositions.
Finally, we presented a general framework for the sensitivity analysis of eigenvalues of matrix pencils and matrix polynomials which lay bare the big picture that lies behind the notion of sensitivity of eigenvalues. We have shown that our treatment unifies various measures of the sensitivity of simple eigenvalues of matrix pencils and matrix polynomials proposed in the literature.
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