Diagonal pencils

In particular, if kxk₂ = 1 = kyk₂ and cos(θ) := |hx, yi| for 0 ≤ θ ≤ π/2, then we have

q(w_max) = 2 cos(θ/2) and q(w_min) = 2 sin(θ/2).

Proof: The desired results follow from the fact that q(w) = kx−wyk₂ =

q

kxk²₂+kyk²₂−2Re(hx, yiw). ¥

Next, we define the notion of departure, denoted by Dep(x, y), between points x and y inC². We define Dep :C² →R by

Dep(x, y) := |y^TJx|

pkxk²₂+kyk²₂+ 2|hx, yi|, (2.5)

whereJ :=

·0 −1 1 0

¸

and y^T is the transpose ofy. It follows that

Dep(x, y)≤ |y^TJx|

kx−wyk₂ (2.6)

forw∈T. If xand y are normalized unit vectors in C² then we have Dep(x, y) = 1

2|y^TJx| sec(θ/2),

where 0 ≤ θ ≤ π/2 and is given by cos(θ) := |hx, yi|. For x := (α₁, β₁) ∈ C² and y:= (α₂, β₂)∈C²,we also write Dep(x, y) as Dep(α₁, β₁;α₂, β₂).In such a case, we have

Dep(α₁, β₁;α₂, β₂) = |α₁β₂−α₂β₁|

pk(α₁, β₁)k²₂+k(α₂, β₂)k²+ 2|h(α₁, β₁),(α₂, β₂)i|

and for normalized (α₁, β₁) and (α₂, β₂), Dep(α1, β1;α2, β2) = 1

2|α1β2−α2β1| sec(θ/2).

For ready reference, we list some essential properties of Dep(x, y).

• Obviously, Dep(x, y) is symmetric, that is, Dep(x, y) = Dep(y, x).

• We have Dep(x, y) = 0 ⇐⇒ x = y in CP¹, the complex project line, that is, x=ty for some t∈C. Equivalently, Dep(α₁, β₁;, α₂, β₂) = 0 ⇐⇒ α₁/β₁ =α₂/β₂ inC_∞:=C∪ {∞}.

• We endowC_∞ with chordal metric as follows: For (α₁, β₁),(α₂, β₂)∈C², chord(β₁/α₁, β₂/α₂) := |α₁β₂−α₂β₁|

k(α1, β1)k2k(α2, β2)k2

.

Then we have

Dep(α₁, β₁;α₂, β₂)≤ 1

√2

pk(α₁, β₁)k₂k(α₂, β₂)k₂chord(β₁/α₁, β₂/α₂).

In particular, for (α₁, β₁),(α₂, β₂)∈S¹ :={x∈C² :kxk₂ = 1}, we have Dep(α₁, β₁;α₂, β₂) = 1

2chord(β₁/α₁, β₂/α₂) sec(θ/2)≤ 1

√2chord(β₁/α₁, β₂/α₂), whereθ ∈[0, π/2] is given by cos(θ) :=|α₁α₂+β₁β₂|.

• We have Dep(tx, ty) =|t|Dep(x, y) for t∈C.

We have already demonstrated that weighted norm on the space of pencils present no extra difficulty in obtaining results. Therefore, for simplicity, we consider w := (1,1) and denote the L²_w(C^n×n,k · k2) by L²(C^n×n,k · k2). Also, we denote the corresponding norm |||·|||_w,2 on L²(C^n×n,k · k₂) be |||·|||₂. Note that the perturbation analysis of pencils inL²(C^n×n,k · k₂) then corresponds to absolute perturbations of matrix pencils.

Let L ∈ L²(C^n×n,k · k₂) be a regular pencil having n distinct eigenvalues. We refer such a pencil as simple pencil. Since d(L) is invariant under unitary equivalence transformation ofL,without loss of generality, we assume thatLis diagonal and is given byL(z) := diag(α_i)−λdiag(β_i) or L(c, s) :=cdiag(α_i)−sdiag(β_i).

Proposition 2.6.4. Let L ∈ L²(C^n×n,k · k₂) be a regular pencil given by L(λ) :=

diag(α_i)−λdiag(β_i). Set x_j := (β_j, α_j) for j = 1 : n. Then Λ_²(L) =∪ⁿ_j=1Λ_²(x_j), where Λ_²(x_j) := {z ∈ C: |α_j −zβ_j| ≤²k(1, z)k₂}. Further, Λ_²(x_i)∩Λ_²(x_j) = ∅ if and only if

² <Dep(x_i, x_j). If ²= Dep(x_i, x_j) then z_w := α_i+w α_j

β_i+w β_j is a common boundary point of Λ_²(x_i) and Λ_²(x_j), where w:= sign(hx_j, x_ii).

Next, consider the homogeneous form of L, that is, L(c, s) =cdiag(α_i)−sdiag(β_i).

Then we have Λ_²(L) = ∪ⁿ_j=1Λ_²(x_j), where Λ_²(x_j) := {(c, s) ∈ S¹ : |c α_j −s β_j| ≤ ²}.

Further, Λ_²(x_i)∩Λ_²(x_j) = ∅ if and only if ² < Dep(x_i, x_j). If ² = Dep(x_i, x_j) then (c_w, s_w) is a common boundary point of Λ_²(x_i) and Λ_²(x_j), where

c_w := β_i+wβ_j

kx_i+wx_jk₂ and s_w := α_i+wα_j kx_i+wx_jk₂. Proof: The fact that Λ_²(L) =∪ⁿ_j=1Λ_²(x_j) is easy to check.

Next, note that if Λ²(xi)∩Λ²(xj) 6= ∅ then there is a complex number z ∈ C such that|αi−zβi|=|αj−zβj|.Hence αi−zβi =t(αj−zβj) for somet∈T. Consequently, we have z = α_i −tα_j

βi −tβj

. Then by (2.6) we have

²= |αi−zβi|

k(1, z)k₂ = |αiβj −αjβi|

kx_i −tx_jk₂ ≥Dep(xi, xj).

On the other hand, we have |αi −zwβi| = |αj −zwβj| and |αi−zwβi|

k(1, z_w)k₂ = Dep(xi, xj).

Hence the results follow. The proof is similar for homogeneous pencils. ¥ This shows that if L(z) := diag(αi)−zdiag(βj) is a simple pencil then

d(L)≥min

i6=j Dep(x_i, x_j).

The following result shows that the equality holds.

Theorem 2.6.5. Let L∈L²(C^n×n,k · k2) be a simple pencil given byL(z) := diag(αi)− zdiag(βi).Setxj := (βj, αj) forj = 1 :n. Then d(L) = mini6=jDep(xi, xj). Suppose that Dep(xi, xj) = mink6=lDep(xk, xl). Let w:= sign(hxj, xii) and zw := α_i+w α_j

β_i+w β_j.

Set U := [e_i, e_j] and V := [sign(α_i−z_wβ_i)e_i, sign(α_j −z_wβ_j)e_j], where e_j is the j-th column of the identity matrix of size n. Define

∆A:=−Dep(x_i, x_j)

k(1, z_w)k₂ UV^∗ and ∆B := z_wDep(x_i, x_j) k(1, z_w)k₂ UV^∗

and consider the pencil ∆L(z) := ∆A−z∆B. Then we have |||∆L|||₂ = d(L) and z_w is a multiple eigenvalue of the pencil L+ ∆L of geometric multiplicity 2.

Proof: Since |α_i−z_wβ_i|=|α_j−z_wβ_j|= Dep(x_i, x_j)k(1, z_w)k₂, it is easy to check that (L(z_w) + ∆L(z_w))V = 0 andU^∗(L(z_w) + ∆L(z_w)) = 0.Hencez_w is a multiple eigenvalue ofL+ ∆L. Consequently, we have |||∆L|||₂ = Dep(x_i, x_j)≥d(L).On the other hand, by Proposition 2.6.4 we have d(L)≥Dep(x_i, x_j). Hence the desired result follows. ¥

The above construction just provides a nearest pencil having a multiple eigenvalue.

We now show how to construct a pencil ∆L such that L+ ∆L is defective and that

|||∆L|||₂ = d(L). We proceed as follows. Recall from Theorem 2.6.5 that L(z) :=

diag(α_i)−zdiag(β_i) is a simple pencil, x_i := (β_i, α_i), x_j := (β_j, α_j), w := sign(hx_j, x_ii), z_w := α_i+wα_j

β_i+wβ_j and d(L) = Dep(x_i, x_j). Then we have

|αi−zwβi|=|αj −zwβj|= Dep(xi, xj)k(1, zw)k2

and

k(1, z_w)k₂ =

pkx_ik²₂+kx_jk²₂+ 2|hx_i, x_ji|

|β_i+wβ_j| = kx_i+wx_jk₂

|β_i+wβ_j| .

Note that α_i −z_wβ_i = −w(α_j −z_wβ_j). Hence sign(α_i −z_wβ_i) = −w sign(α_j −z_wβ_j).

Note also that ek and eksign(αk − zwβk) are left and right singular vectors of L(zw) corresponding to |αk−zwβk|,for k =i, j. For t∈(0,1), we define

u:=te_i+√

1−t²e_j and v := (te_i−√

1−t²we_j)sign(α_i−z_wβ_i).

Then it is easily checked that u and v are unit left and right singular vectors of L(z_w) corresponding to |α_i−z_wβ_i|. Now we determine t such that u and v satisfies

u^∗Bv+z_w|α_i−z_wβ_i| k(1, zw)k²₂ = 0.

This gives t²β_i−(1−t²)wβ_j+ z_w(α_i−z_wβ_i)

k(1, z_w)k²₂ = 0. Consequently, we have t²(βi+wβj) = k(1, zw)k²₂(βi+wβj)−(βi+zwαi)

k(1, z_w)k²₂ = (βi+wβj)− kxik²₂+|hxi, xji|

(β_i+wβ_j)k(1, z_w)k²₂ which gives

t² := kx_jk²₂+|hx_i, x_ji|

kx_i+wx_jk²₂ and 1−t² := kx_ik²₂+|hx_i, x_ji|

kx_i+wx_jk²₂ . Thus, we have the following result.

Theorem 2.6.6. Let L∈L²(C^n×n,k · k₂) be a simple pencil given by L(z) = diag(α_i)− zdiag(β_i). Let x_i, x_j, w and z_w be as in Theorem 2.6.5. Set

t :=

pkx_jk²₂+|hx_i, x_ji|

kx_i+wx_jk₂ and define u := te_i+√

1−t² e_j and v := (te_i−√

1−t² we_j)sign(α_i−z_wβ_i). Further, define

∆A:=−Dep(x_i, x_j) k(1, zw)k2

uv^∗ and ∆B := z_wDep(x_i, x_j) k(1, zw)k2

uv^∗

and consider the pencil ∆L(z) := ∆A−z∆B. Then |||∆L|||2 = d(L) = Dep(xi, xj) and z_w is a defective eigenvalue of the pencil L+ ∆L.

Proof: Note that by constructionuandvare unit left and right singular vectors ofL(z_w) corresponding to σ :=|α_i−z_wβ_i|. Also, by construction u and v are unit left and right eigenvectors ofL+∆Lcorresponding to the eigenvaluez_w.Sinceu^∗Bv+ z_wσ

k(1, zw)k²₂ = 0, by Theorem 2.4.11, z_w is a multiple eigenvalue of L + ∆L. Note that rank(L(z_w) +

∆L(z_w)) =n−1. Consequently, z_w is a defective eigenvalue L+ ∆L. By construction, we have |||∆L|||₂ = Dep(x_i, x_j). Hence the proof. ¥

For completeness, we now derive the homogeneous version of Theorem 2.6.6.

Theorem 2.6.7. LetL ∈L²(C^n×n,k·k2)be a simple pencil given byL(c, s) =cdiag(αi)−

sdiag(βi). Set xj := (βj, αj) for j := 1 :n. Then we have d(L) = min

i6=j Dep(x_i, x_j).

Let xi and xj be such that Dep(xi, xj) = d(L). Set w:= sign(hxj, xii) and define cw := βi+wβj

kx_i+wx_jk₂ and sw := αi+wαj

kx_i+wx_jk₂.

Then (c_w, s_w)∈S¹ and |c_wα_i−s_wβ_i|=|c_wα_j −s_wβ_j|= Dep(x_i, x_j). Next, set t :=

pkx_jk²₂+|hx_i, x_ji|

kx_i+wx_jk₂ and define u:=tei+√

1−t² ej and v := (tei−√

1−t² wej)sign(cwαi−swβi). Further, define

∆A:=−c_wDep(x_i, x_j)uv^∗ and ∆B :=s_wDep(x_i, x_j)uv^∗

and consider the pencil∆L(c, s) :=c∆A−s∆B. Then |||∆L|||₂ = d(L) and (c_w, s_w) is a defective eigenvalue of the pencil L+ ∆L.

Proof: By construction, we have

|c_w|²+|s_w|² = |β_i+wβ_j|²+|α_i+wα_j|² kx_i+wx_jk²₂ = 1.

This shows that (c_w, s_w)∈S¹. Now c_wα_i−s_wβ_i = β_i+wβ_j

k(x_i+wx_j)k₂α_i− α_i+wα_j

k(x_i+wx_j)k₂β_i = w(α_iβ_j −α_jβ_i) kx_i+wx_jk₂ . Consequently, we have

|cwαi−swβi|= |α_jβ_i−α_iβ_j|

k(x_i+wx_j)k₂ = |α_jβ_i−α_iβ_j|

p(kx_ik²+kx_jk²+ 2| hx_i, x_ji |) = Dep(xi, xj).

Similarly, we have that|cwαj −swβj|= Dep(xi, xj). Hence

|cwαi−swβi|=|cwαj −swβj|= Dep(xi, xj).

By construction, u and v are unit left and right singular vectors of L(c_w, s_w) cor- responding to the smallest singular value Dep(x_i, x_j). Also, by construction, we have (L(c_w, s_w) + ∆L(c_w, s_w))v = 0 and u^∗(L(c_w, s_w) + ∆L(c_w, s_w)) = 0. We now show that

u^∗Av−c_wDep(x_i, x_j) = 0 and u^∗Bv+s_wDep(x_i, x_j) = 0.

Now, we have

u^∗Av−c_wDep(x_i, x_j) =

·

t²(α_i+wα_j)−wα_j

¸

sign(c_wα_i−s_wβ_i)−c_w|c_wα_i−s_wβ_i|

= sign(c_wα_i−s_wβ_i)

·

[t²(α_i+wα_j)−wα_j]−c_w(c_wα_i−s_wβ_i)

¸ . Since|c_w|²+|s_w|² = 1,we have [t²(α_i+wα_j)−wα_j]−c_w(c_wα_i−s_wβ_i) = t²(α_i+wα_j)− (α_i+wα_j) +s_w(s_wα_i +c_wβ_i) = (α_i+wα_j)

·

t² −1 + kx_ik²₂+whx_j, x_ii kx_i+wx_jk²₂

¸

= 0.

This shows thatu^∗Av−c_wDep(x_i, x_j) = 0.Similarly, we haveu^∗Bv+s_wDep(x_i, x_j) = 0. Hence by Theorem 2.4.11, (c_w, s_w) is a multiple eigenvalue of L+ ∆L. Note that rank(L(c_w, s_w) + ∆L(c_w, s_w)) = n −1. Hence (c_w, s_w) is a defective eigenvalue. This completes the proof. ¥

We mention that it is easy to derive analogues of the above results when L ∈ L²_w(C^n×n,k · k₂). Indeed, define the weighted scalar product on C² by

hx, yi_w :=w²₁x₁y₁+w²₂x₂y₂ forx, y ∈C². Then|hx, yi_w| ≤ kxk_w,2kyk_w,2.Now for x, y ∈C², define

Dep_w(x, y) := |x₂y₁−x₁y₂|

(kxk²_w−1,2+kyk²_w−1,2+ 2|hx, yi_w⁻¹|)^1/2. Then we have the following result.

Theorem 2.6.8. LetL ∈L²_w(C^n×n,k·k2)be a simple pencil given byL(c, s) =cdiag(αi)−

sdiag(βi). Set xj := (βj, αj) for j := 1 :n. Then we have d(L) = min

i6=j Dep_w(xi, xj).

Let x_i and x_j be such that Dep_w(x_i, x_j) = d(L). Set e:= sign(hx_j, x_ii_w⁻¹) and define c_e:= β_i+eβ_j

kxi+exjk_w⁻¹_,2 and s_e := α_i+eα_j kxi+exjk_w⁻¹_,2.

Then (c_e, s_e)∈S¹ and |c_eα_i−s_eβ_i|=|c_eα_j −s_eβ_j|= Dep_w(x_i, x_j). Next, set t:=

q

kx_jk²_w−1,2+|hx_i, x_ji_w⁻¹| kxi+exjk_w⁻¹_,2 and define u :=te_i +√

1−t² e_j and v := (te_i −√

1−t² ee_j)sign(c_eα_i−s_eβ_i). Further, define

∆A:=−c_ew⁻²₁ Dep(x_i, x_j)

k(c_e, s_e)k_w⁻¹_,2 uv^∗ and ∆B := s_ew₂⁻²Dep(x_i, x_j) k(c_e, s_e)k_w⁻¹_,2 uv^∗

and consider the pencil ∆L(c, s) := c∆A−s∆B. Then |||∆L|||_w,2 = d(L) and (c_e, s_e) is a defective eigenvalue of the pencil L+ ∆L.

We now illustrate Wilkinson’s problem by considering a few simple examples. For simplicity, we assume that w:= (1,1) and denote η_w,2(z,L) by η(z,L).

Example 2.6.9. Let L(z) :=A−zB be a diagonal pencil given by (a) A :=

·−1 +i 0 0 1 + 2i

¸

, B :=

·1/2 0

0 1

¸

(b) A :=

·1 + 2i 0 0 2 +i

¸

, B :=

·1 0 0 2

¸ .

Then for (a) we have d(L) := 0.3878, z_w := 0.1628 + 2.7905i and for (b) we have d(L) := 0.5628, z_w := 1.1564 + 0.9877i.Figure 2.2 shows the coalescence of components of Λ_²(L). ¥

−10 −5 0

0 2 4 6 8 10

X−axis

Y−axis

Zw

A=([−1+i;1+2i]) , B=diag([1/2; 1]), ε₀=0.3878

λ₁ λ₂

0 1 2 3

0 1 2 3 4 5

X−axis

Y−axis λ₁

λ₂

A=diag([1+2i; 2+i]), B=diag([1; 2]) ,

Zw

ε₀=0.5627

Figure 2.2: The left and right figures show coalescence of components of Λ²(L) corre- sponding to the pencil given in (a) and (b), respectively.

The pseudospectra plot of a matrix pencilL in C could be quite difficult to analyze when L has an infinite eigenvalue or when all the eigenvalues of L are finite but “infin- ity” enter into the pseudospectrum before the coalescence of two components. This is illustrated by the following example.

Example 2.6.10. Consider L(z) :=A−zB,where A:=

·−0.2 + 0.3i 0 0 0.3−0.2i

¸ and B :=

·0.1 0 0 0.2

¸

.The eigenvalues of L are given byλ₁ :=−2 + 3i and λ₂ := 3/2−i. By Theorem 2.6.6, we have d(L) := 0.145278 and z_w := 5.9567−1.0175i. Figure 2.3 shows the contour plots of Λ_²(L) and the point of coalescence of components. The evolution of the components of Λ_²(L) is quite interesting in this case. Note that for sufficiently small

²,the components of Λ_²(L),are bounded regions inCcontaining the eigenvalues λ₁ and λ₂ in their interiors. As ² grows gradually to 0.145278, the component containing λ₂ remains trapped in a bounded region of the complex plane. By contrast, as²→0.145278, the component containingλ1expands very fast engulfing the entire complex plane except for a small bounded region before coalescing with the component containing λ₂ at z_w. Indeed, for²:= 0.145278,the pseudospectrum Λ_²(L) is multiply connected and consists of entire complex plane except for the shaded region. Note that ∞ enters into the component of Λ_²(L) containingλ₁ before it coalesces with the component containingλ₂ at z_w for ² := 0.145278. The dynamics of the evolution of the components of Λ_²(L) is

better captured by the surface plot of the map (x, y)7→η(x+iy,L). Figure 2.4 clearly shows the evolution of the components of Λ_²(L). ¥

−6 −4 −2 0 2 4 6

−6

−4

−2 0 2 4 6

X−axis

Y−axis

A=diag([−0.2+0.3i;0.3−0.2i]),B = diag([0.1;0.2]),ε₀=0.145278

Zw

λ₁

λ₂

Figure 2.3: Contour plots of Λ_²(L). The components coalesce at z_w for ² := 0.145278.

For ² := 0.145278, Λ²(L) is multiply connected and consists of entire complex plane except for the shaded region.

For cases such as this, contour plots of Λ²(L) in the complex plane is inefficient and can be potentially confusing if not interpreted properly. The best way to overcome such problem as well as the case whenLhas an infinite eigenvalue is to consider homogeneous form ofLand plot Λ_²(L) on the Riemann sphere S:={(x, y, z)∈R³ :x²+y²+z² = 1}.

Indeed, by considering the homogeneous pencil L(c, s) = cA−sB, we can restrict c to be real and s to be complex. Then the north pole (0,0,1) of S represents ∞. Note that each point (x, y, z) ∈ S, except the north pole (0,0,1), can be associated with a complex number w:= x−iy

1 −z. It is easily seen that this correspondence is one-to-one and x= w +w

1 +|w|², y = w− w

i(1 +|w|²). Further, it follows that the eigenvalue 0 of L is represented by the south pole (0,0,−1) of S.

Now consider once again the pencilLgiven in Example 2.6.10. Considering homoge- neous form ofL,Figure 2.5 shows the plot of Λ_²(L) on the Riemann sphereS.Note that as² grows gradually, the component of Λ_²(L) containing the (unnormalized) eigenvalue (0.1, −0.2 + 0.3i) expands and contains (0,0,1) before coalescing with the component containing the (unnormalized) eigenvalue (0.2, 0.3−0.2i) at (c_w, s_w) for ²:= 0.145278, where the point (c_w, s_w) is given by Theorem 2.6.7.

Finally, we consider an example of a pencil for which∞ is an eigenvalue.

−4 −2 0 2 4 6

−5 0

5 0 0.5 1 1.5 2 2.5 3

X−axis Surface Plot

Y−axis

Z−axis

Figure 2.4: Surface plot of the map (x, y) 7→ η(x+iy,L) showing the evolution of components of Λ_²(L).

Example 2.6.11. Consider the pencil L(z) = A−zB, where A := diag(3,1 +i) and B := diag(0,2). Figure 2.6 shows the plot of Λ²(L) on the Riemann sphere S and the coalescence of the components of Λ²(L) for ² := 1.2381. By Theorem 2.6.7, we have d(L) := 1.2381. ¥

Figure 2.5: Plot of Λ_²(L) on the Riemann sphere S for the pencil L given in Exam- ple 2.6.10 showing the coalescence of components.

Figure 2.6: Plot of Λ_²(L) on S showing coalescence of components.

Chapter 3

On pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for defining and analyzing pseudospectra of matrix polynomials. The framework so developed parallels the well developed framework for pseudospectra of matrices. We show that the pseudospectra of matrix polynomials well known in the literature follow as special cases from our framework. We also analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. Finally, we show that common boundary points of the components of pseudospectra of a matrix polynomial are critical points. In particular, we show that a minimal critical point can be read off from the pseudospectra of matrix polynomials.

Hence a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.

3.1 Introduction

We further developed the framework introduced for defining and analyzing pseudospectra of matrix pencils to the case of matrix polynomials. We define the pseudospectra of regular matrix polynomials in the general framework of normed/seminormed linear space of matrix polynomials. We show that our framework for analyzing pseudospectra of matrix polynomials parallels the well developed framework that exists for analyzing pseudospectra of matrices. The crux of the matter is that, like the pseudospectra of matrices, the pseudospectra of matrix polynomials are determined by the geometry of the normed/seminormed space of polynomials, that is, by the choice of the norm/seminorm on the space of polynomials. We show that the pseudospectra of matrix polynomials well known in the literature correspond to appropriate choices of norm/seminorm on the

space of polynomials.

Next, we consider critical points of backward errors of approximate eigenvalues of matrix polynomials and analyze their significance. We equip the space of polynomials with a weighted H¨older’s p-norm/seminorm |||·|||w,p (defined in section 3.3). Given a regular matrix polynomial L(z) = P_m

i=0zⁱAi and λ ∈ C, we denote by ηw,p(λ,L) the backward error ofλ as an approximate eigenvalue of L,that is,

η_w,p(λ,L) := inf{|||∆L|||_w,p : det(L(λ) + ∆L(λ)) = 0}.

Then we show that ifλis a critical point ofη_w,p then there exists a polynomial ∆Lsuch that |||∆L|||_w,p =η_w,p(λ,L) and thatλ is a multiple eigenvalue of L+ ∆L.Indeed, when λ is a generic critical point of η_w,p,setting

∆A_i :=−η_w,p(λ,L) (∇_iN)(1, λ, . . . , λ^m)uv^∗ and defining ∆L(z) := P_m

i=0zⁱ∆Ai we obtain the desired polynomial, where u and v, respectively, are normalized left and right singular vectors of L(λ) corresponding to the smallest singular value σ_min(L(λ)), N(z₁, z₂, . . . , z_m) := k(z₁, z₂, . . . , z_m)k_w⁻¹_,q, (∇_iN)(1, λ, . . . , λ^m) is the partial gradient of N evaluated at (1, λ, . . . , λ^m) and p⁻¹ + q⁻¹ = 1.Whenλis a nongeneric critical point ofη_w,p,we show that a similar construction for ∆L holds. Thus we show that if λ is a critical point (either generic or nongeneric) of η_w,p then λ is a multiple eigenvalue of a polynomial which lies on the boundary of the η_w,p(λ,L)-neighbourhood of L. We show that certain critical points of η_w,p can be read off from the pseudospectra ofL.More specifically, we show that common boundary points of the components of pseudospectra ofL are critical points ofηw,p. In particular, when L is simple (that is, it has distinct eigenvalues) we show that a minimal critical point of ηw,p can be read off from the pseudospectra of L. Hence we show that the distance from L to the nearest polynomial having a multiple eigenvalue can be read off from the pseudospectra ofL. Thus we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of the polynomials.