Structured backward error

Recall that the spectrum of a regular polynomial P∈Pm(C^n×n) is denoted byσ(P),(chap- ter 1). In this chapter, we consider only finite eigenvalues of matrix polynomials. By conven- tion, if (λ, x) ∈C×Cⁿ thenx is assumed to be nonzero, that is,x6= 0. Treating (λ, x) as an approximate eigenpair of the polynomial P∈P_m(C^n×n),we define the backward error of (λ, x) by

η_F(λ, x,P) := inf

4P∈Pm(C^n×n){|||4P|||_F : P(λ)x+4P(λ)x= 0}

η2(λ, x,P) := inf

4P∈P_m(C^n×n){|||4P|||2: P(λ)x+4P(λ)x= 0}.

Further, for (λ, x)∈C×Cⁿ,settingr:=−P(λ)x,we have ηF(λ, x,P) = krk2

kxk2kΛmk2 =η2(λ, x,P). (4.1) Indeed, defining 4Aj := λ^jrx^H

x^HxkΛmk²₂, j = 0 : m, and considering the polynomial 4P(z) = P_m

j=0z^j4Aj, we have |||4P|||F = krk2/kxk2kΛmk2 = |||4P|||2 and P(λ)x+4P(λ)x = 0.

Henceforth, we denote the unstructured backward error with respect to both Frobenius and spectral norms byη(λ, x,P).

Next assume that P∈S.Then treating (λ, x)∈C×C^n×n as an approximate eigenpair of P,we define the structured backward error of (λ, x) by

η_F^S(λ, x,P) := inf

4P∈S{|||4P|||F : P(λ)x+4P(λ)x= 0}

η₂^S(λ, x,P) := inf

4P∈S{|||4P|||2: P(λ)x+4P(λ)x= 0}.

Henceforth, we denote the structured backward error with respect to both Frobenius and spec- tral norms byη^S(λ, x,P). Obviously, we have η(λ, x,P)≤η^S(λ, x,P) and by Theorem 4.2.1, η^S(λ, x,P)<∞.

Now we derive the structured backward error of an approximate eigenpair (λ, x) of struc- tured matrix polynomials. Recall that for (λ, x)∈C×Cⁿ, our standing assumption is that

x^Hx= 1.First, considerT-symmetric matrix polynomials. Recall that Λm:= [1, λ, . . . , λ^m]^T. Theorem 4.3.1. Let S be the space of T-symmetric matrix polynomials andP∈S be given by P(z) =P_m

j=0z^jAj. Then for (λ, x)∈C×Cⁿ, settingr:=−P(λ)x,we have η_F^S(λ, x,P) =

p2krk²₂− |x^Tr|² kΛmk2 ≤√

2η(λ, x,P), η₂^S(λ, x,P) =η(λ, x,P).

Now define 4Aj := _kΛ^λj

mk²₂[xr^T +rx^H−(r^Tx)xx^H], j = 0 :m and consider the polynomial 4P(z) = P_m

j=0z^j4Aj. Then 4P is a unique polynomial such that 4P ∈ S, 4P(λ)x+ P(λ)x= 0and|||4P|||F =η^S_F(λ, x,P). Further, define

4Aj:= λ^j

kΛmk²₂[xr^T +rx^H−(r^Tx)xx^H]− λ^j x^Tr P_x^Trr^TPx

kΛmk²₂ (krk²₂− |x^Tr|²) and consider the polynomial4P(z) =P_m

j=0z^j4Aj.Then4P∈S,4P(λ)x+ P(λ)x= 0 and

|||4P|||2=η^S₂(λ, x,P).

Proof: By Theorem 4.2.1 there exists 4P ∈ S such that P(λ)x+4P(λ)x = 0. Then we have r=4P(λ)x.ChooseQ1 ∈C^n×(n−1) such that the matrixQ= [x Q1] is unitary. Let 4A]j :=Q^T4AjQ =

Ã

ajj a^T_j aj Xj

!

, where Xj =X_j^T is of size n−1. Since QQ^T =I, we have

Q(4P(λ))Q^Hx=r⇒(4P(λ))Q^Hx=Q^Tr= Ã x^Tr

Q^T₁r

!

AsQ^Hx=e1,the first column of the identity matrix, we have à P_m

j=0λ^jajj

P_m

j=0λ^jaj

!

= Ã x^Tr

Q^T₁r

!

whose minimum solutions are aj= ^λ_kΛ^jQT1r

mk²₂, ajj = _kΛ^λjxTr

mk²₂, j= 0 :m.Hence we have 4Ag_j =





λ^jx^Tr

kΛmk²₂ (^λ_kΛ^jQT1r

mk²₂)^T

λ^jQ^T₁r

kΛmk²₂ X_j



. (4.2)

This shows that the Frobenius norm of 4A]j’s are minimized whenXj = 0. Hence we have k4Ajk²_F =k4A]jk²_F =|ajj|²+ 2kajk²₂.SinceQ₁Q^T₁ =I−xx^T,we have

η^S_F(λ, x,P) = s

|x^Tr|²

kΛmk²₂ +2k(I−xx^T)rk²₂ kΛmk²₂ =

p2krk²₂− |x^Tr|² kΛmk2 .

Simplifying the expressions of4Aj we have

4Aj = [x Q₁]





λ^jx^Tr

kΛmk²₂ (^λ_kΛ^jQT1r

mk²₂)^T

λ^jQ^T₁r kΛmk²₂ 0



 Ã

x^H Q^H₁

!

= λ^j

kΛmk²₂(x^Tr)xx^H+ λ^j

kΛmk²₂xr^TQ1Q^H₁ + λ^j

kΛmk²₂Q₁Q^T₁rx^H

= λ^j

kΛmk²₂[xx^Trx^H+xr^T(I−xx^H) + (I−xx^T)rx^H]

= λ^j

kΛmk²₂[xr^T+rx^H−(r^Tx)xx^H] from which we obtain the desired polynomial 4P.

From (4.2) considerµ4Aj = ^|λ_kΛ^{j| krk2}

mk²₂ .Then by DKW Theorem 1.2.5 we have, 4Xj=− λ^j x^Tr Q^T₁r(Q^T₁r)^T

kΛmk²₂ (krk²₂− |x^Tr|²), j= 0 :m, which gives,

η₂^S(λ, x,P) = krk2

kΛmk2 =η(λ, x,P).

Simplifying the expression of4Aj we obtain 4Aj= λ^j

kΛmk²₂[xr^T +rx^H−(r^Tx)xx^H]− λ^j x^Tr P_x^Trr^TPx

kΛmk²₂ (krk²₂− |x^Tr|²). This completes the proof.¥

Remark 4.3.2. If |x^Tr|=krk2,then kQ^T₁rk2= 0. In such a case, considering Xj = 0, j = 0 :m, we obtain the desired results.

Observe that ifY is symmetric andY x= 0 thenY =P_x^TZPxfor some symmetric matrix Z.Consequently, we have Q_jX_jQ^H_j =P_x^TZ_jP_x, j = 0 :m, for some symmetric matrices Z_j. Hence from the proof of Theorem 4.3.1 we have following.

Corollary 4.3.3. Let P(z) = P_m

j=0z^jA_j be a T-symmetric polynomial and let (λ, x) ∈ C×Cⁿ.Set r=−P(λ)x.ThenP(λ)x+ Q(λ)x= 0 if and only ifQ(z) =4P(z) +P_x^TR(z)Px

for some T-symmetric polynomial R ∈ P_m(C^n×n), where 4P(z) = P_m

j=0z^j4A_j is the T- symmetric polynomial, given by

4Aj := λ^j

kΛmk²₂[xr^T +rx^H−(r^Tx)xx^H] Next we considerT-skew-symmetric polynomials.

Theorem 4.3.4. Let S be the space of T-skew-symmetric matrix polynomials andP∈S be given by P(z) =P_m

j=0z^jAj. Then for (λ, x)∈C×Cⁿ, settingr=−P(λ)x,we have η_F^S(λ, x,P) =√

2 η(λ, x,P), η₂^S(λ, x,P) =η(λ, x,P).

Further, for the T-skew-symmetric polynomial 4P given in Theorem 4.2.1 we have (P(λ) + 4P(λ))x= 0,|||4P|||F =η^S_F(λ, x,P)and|||4P|||2=η^S₂(λ, x,P).

Proof: The arguments proceed on the lines as those given in the proof of Theorem 4.3.1.

The only difference is the fact that, in this case, 4Aj is skew-symmetric for all j = 0 : m.

Thus we have

4A]j:=Q^T4AjQ=

à 0 a^T_j

−aj Xj

!

, Q^Tr= Ã 0

Q^T₁r

!

, X_j^T =−Xj. Consequently, we have

à 0

−P_m

j=0λ^jaj

!

= Ã x^Tr

Q^T₁r

!

⇒x^Tr= 0, aj=−λ^jQ^T₁r kΛmk²₂. Hence we have

4Aj=Q



 0 −(^λ_kΛ^jQT1r

mk²₂)^T

λ^jQ^T₁r

kΛmk²₂ X_j



Q^H. (4.3)

SettingXj= 0,we obtain the polynomial4P such that|||4P|||F =η^S_F(λ, x,P) =√

2krk2/kΛmk2. Next, sinceQ₁Q^T₁ = I−xx^T,simplifying the expressions we have4Ai=_kΛ^λj

mk²₂[rx^H−(rx^H)^T], from which theT-skew-symmetric polynomial4P given in Theorem 4.2.1 follows. This com- pletes the proof for Frobenius norm.

Next from (4.3) we have µ4Aj = ^|λ_kΛ^{j| krk2}

mk²₂ . Hence by DKW Theorem 1.2.5, we have, Xj= 0.Thus we obtainη₂^S(λ, x,P) =η(λ, x,P).The desired result follows by simplifying the expression of4Aj.This completes the proof.¥

Using the fact that if Y is skew-symmetric and Y x = 0 then Y = P_x^TZPx for some skew-symmetric matrix Z, we obtain an analogue of Corollary 4.3.3 for T-skew-symmetric polynomials.

Corollary 4.3.5. Let P ∈ Pm(C^n×n) be a T-skew-symmetric polynomial and let (λ, x) ∈ C×Cⁿ.Set r=−P(λ)x.ThenP(λ)x+ Q(λ)x= 0 if and only ifQ(z) =4P(z) +P_x^TR(z)Px

for some T-skew-symmetric polynomial R∈Pm(C^n×n), where 4P is the T-skew-symmetric polynomial given in Theorem 4.3.4.

To describe the structured backward errors forT-even andT-odd polynomials in a con- venient manner, we define the even index projection Πe:C^m+1→C^m+1 by

Πe([x0, x1, x1, . . . , xm−1, xm]^T) :=

( [x0, 0, x2, 0, . . . , xm−2,0, xm]^T, ifmis even, [x₀, 0, x₂, 0, . . . ,0, x_m−1,0]^T, ifmis odd.

Note that “0” is considered as even number. ThenI−Πe is the odd index projection.

Theorem 4.3.6. Let S be the space of T-even matrix polynomials and P ∈ S be given by P(z) =P_m

j=0z^jAj.Then for (λ, x)∈C×Cⁿ,setting r=−P(λ)x,we have η_F^S(λ, x,P) =

s

|x^Tr|²

kΠe(Λm)k²₂ + 2krk²₂− |x^Tr|²

kΛmk²₂ , η₂^S(λ, x,P) = s

|x^Tr|²

kΠe(Λm)k²₂ +krk²₂− |x^Tr|² kΛmk²₂ .

In particular, if m is odd and |λ| = 1, then η_F^S(λ, x,P) = √

2 η(λ, x,P), η₂^S(λ, x,P) = η(λ, x,P).

Let

Ej:=











λ^j

kΠe(Λm)k²₂(x^Tr)xx^H+ λ^j

kΛmk²₂[xr^TPx+P_x^Trx^H], ifj is even λ^j

kΛmk²₂[P_x^Trx^H−xr^TPx], if j is odd . Setting 4Aj = Ej we obtain a unique T-even polynomial 4P(z) = P_m

j=0z^j4Aj such that P(λ)x+4P(λ)x= 0 and|||4P|||F =η_F^S(λ, x,P).Further, for j= 0 :mdefining

4Aj:=







Ej− λ^j x^Tr P_x^Trr^TPx

kΠ_e(Λ_m)k²₂ (krk²₂− |x^Tr|²), ifj is even

Ej, ifj is odd

we obtain a T-even polynomial 4P(z) = P_m

j=0z^j4Aj such that P(λ)x+4P(λ)x= 0 and

|||4P|||₂=η^S₂(λ, x,P).

Proof: Suppose S is a space of T-even polynomials and P(z) = P_m

j=0z^jAj ∈ S. Note that A_j is symmetric when j is even ( including “0”) and skew-symmetric when j is odd.

The proof follows from similar arguments as those employed for T-symmetric and T-skew- symmetric polynomials. Indeed, considering a unitary matrix Q:= [x, Q1], we have4Aj = Q

à ajj a^T_j aj Xj

!

Q^H, X_j^T =Xj,ifj is even, and4Aj =Q

à 0 b^T_j

−bj Yj

!

Q^H, Y_j^T =−Yj, ifj is odd.

Consequently we have

à P

jλ^jajj

P

j-evenλ^jaj−P

j-oddλ^jbj

!

= Ã

x^Tr Q^T₁r

! .

Hence the smallest norm solutions areajj = _kΠ ^λj

e(Λm)k²₂ x^Tr, aj= _kΛ^λj

mk²₂Q^T₁r, bj=−_kΛ^λj

mk²₂Q^T₁r.

Therefore we have

4Aj=























 Q







λ^j

kΠe(Λm)k²₂x^Tr (^λ_kΛ^jQT1r

mk²₂)^T

λ^jQ^T₁r

kΛmk²₂ Xj





Q^H, ifj is even

Q







0 −(^λ_kΛ^jQT1r

mk²₂)^T

λ^jQ^T₁r

kΛmk²₂ Y_j





Q^H, ifj is odd.

(4.4)

SettingXj= 0 =Yj and using the fact thatQ₁Q^T₁ = I−xx^T,we obtain a unique polynomial 4P(z) =P_m

j=0z^j4Aj such that

|||4P|||F =η^S_F(λ, x,P) = s

|x^Tr|²

kΠe(Λm)k²₂ + 2krk²₂− |x^Tr|² kΛmk²₂ .

Ifmis odd and|λ|= 1 then notice thatkΠe(Λm)k²₂= ¹₂kΛmk²₂.Hence we obtainη_F^S(λ, x,P) =

√2η(λ, x,P). Now simplifying expressions for 4Aj we obtain 4Aj = Ej. Therefore, we obtain a T-even polynomial 4P(z) = P_m

j=0z^j4Aj such that P(λ)x+4P(λ)x = 0 and

|||4P|||F =η_F^S(λ, x,P).

For the spectral norm, we considerµ4A_j =

q|λ^j|²|x^Tr|²

kΠe(Λm)k⁴₂ +^{|λj|2 (krk}_kΛ^22−|xTr|2)

mk⁴₂ ,ifj is even andµ4A_j =

q|λ^j|² (krk²₂−|x^Tr|²)

kΛmk⁴₂ ,ifj is odd.

Then by DKW Theorem 1.2.5 and (4.4) we have, Xj = − λ^j x^Tr Q^T₁r(Q^T₁r)^T

kΠe(Λm)k²₂ (krk²₂− |x^Tr|²) and Yj = 0 which gives,

η^S₂(λ, x,P) = s

|x^Tr|²

kΠe(Λm)k²₂ +krk²₂− |x^Tr|² kΛmk²₂ . Now simplifying expressions for 4Aj we obtain the desired result.¥

The above proof shows that setting4Aj:=Ej+P_x^TZjPx,whereZ_j^T =Zj whenj is even and Z_j^T =−Zj when j is odd, we obtain a T-even polynomial 4P(z) =P_m

j=0z^j4Aj such that P(λ)x+4P(λ)x= 0.

Remark 4.3.7. If |x^Tr|=krk2,then kQ^T₁rk2= 0. In such a case, consideringXj= 0 =Yj

we obtain the desired results.

Next we consider backward error ofT-odd polynomials.

Theorem 4.3.8. Let S be the space of T-odd matrix polynomials and P ∈ S be given by P(z) =P_m

j=0z^jAj.Then for (λ, x)∈C×Cⁿ,setting r=−P(λ)x,we have η_F^S(λ, x,P) =

( q |x^Tr|²

k(I−Πe)(Λm)k²₂ + 2^krk_kΛ^22−|xTr|2

mk²₂ , if λ6= 0

√2η(λ, x,P), if λ= 0

η₂^S(λ, x,P) =

( q |x^Tr|²

k(I−Πe)(Λm)k²₂ +^krk_kΛ^22−|xTr|2

mk²₂ , if λ6= 0

η(λ, x,P), if λ= 0

In particular, if m is odd and |λ| = 1, then η_F^S(λ, x,P) = √

2 η(λ, x,P), η₂^S(λ, x,P) = η(λ, x,P).

Let

F_j:=









 λ^j

kΛmk²₂[P_x^Trx^H−xr^TPx], if j is even λ^jxx^Trx^H

k(I−Πe)(Λm)k²₂ + λ^j

kΛmk²₂[xr^TPx+P_x^Trx^H], if j is odd.

Defining 4Aj :=Fj, we obtain a uniqueT-odd polynomial 4P(z) =P_m

j=0z^j4Aj such that P(λ)x+4P(λ)x= 0 and|||4P|||F =η_F^S(λ, x,P).

Further for the spectral norm, define4Aj :=Fj,if j is even and 4Aj :=Fj− λ^jx^TrP_x^Trr^TPx

k(I−Πe)Λmk²₂(krk²₂− |x^Tr|²),

if j is odd. Then we obtain a T-odd polynomial 4P(z) = P_m

j=0z^j4Aj such that P(λ)x+ 4P(λ)x= 0and |||4P|||2=η^S₂(λ, x,P).

Proof: By interchanging the role ofAj for even and odd j the desired result follows from the proof of Theorem 4.3.6. ¥

The above Theorem shows that setting4Aj =Ej+P_x^TZjPx,where Z_j^T =−Zj whenj is even, and Z_j^T =Zj whenj is odd, we obtain a T-odd polynomial4P(z) = P_m

j=0z^j4Aj

such that P(λ)x+4P(λ)x= 0.

To make the presentation simple we introduce the following operators.

Re:C^m+1→R^m+1 is defined by Re([x0, x1, . . . , xm]^T)7→[re(x0), re(x1), . . . ,re(xm)]^T Im:C^m+1→R^m+1 is defined by Im([x0, x1, . . . , xm]^T)7→[im(x0),im(x1), . . . ,im(xm)]^T. Then forx∈C^m+1 we havex=Re(x) +iIm(x),wherei =√

−1 andre(z),im(z) denote the real and imaginary parts of a complex numberz,respectively.

Theorem 4.3.9. LetSbe the space of H-Hermitian or H-skew-Hermitian matrix polynomials andP∈S be given by P(z) =P_m

j=0z^jAj. Then for(λ, x)∈C×Cⁿ, settingr=−P(λ)x,we have

η^S_F(λ, x,P) =





√2krk²₂−|x^Hr|² kΛmk2 ≤√

2η(λ, x,P), if λ∈R q

kbrk²₂+^2(krk_kΛ^22−|xHr|2)

mk²₂ , if λ∈C\R.

η^S₂(λ, x,P) =

( η(λ, x,q P), if λ∈R kbrk²₂+^krk_kΛ^22−|xHr|2

mk²₂ , if λ∈C\R.

where rb=

"

Re(Λm)^T Im(Λm)^T

#_†"

re(x^Hr) im(x^Hr)

#

for H-Hermitian andrb=

"

−Im(Λm)^T Re(Λm)^T

#_†"

re(x^Hr) im(x^Hr)

# for H-skew-Hermitian polynomial.

Next, let E=xr^H+rx^H−(r^Hx)xx^H andF =rx^H−xr^H+ (r^Hx)xx^H. Whenλ∈R,define

4Aj:=







 λ^j

kΛmk²₂E, ifAj=A^H_j λ^j

kΛmk²₂F, ifAj=−A^H_j and forλ∈C\R,define

4Aj:=







e^T_jbrxx^H+ 1

kΛmk²₂[λ^jPxrx^H+λ^jxr^HPx], if Aj=A^H_j ie^T_jrxxb ^H+ 1

kΛmk²₂[λ^jPxrx^H−λ^jxr^HPx], if Aj=−A^H_j . Now Consider the polynomial 4P(z) = P_m

j=0z^j4Aj. Then 4P ∈ S is unique such that P(λ)x+4P(λ)x= 0 and|||4P|||F =η_F^S(λ, x,P).

Further, forλ∈R, define

4Aj :=







 λ^j

kΛmk²₂E− λ^j x^HrPxrr^HPx

kΛmk²₂ (krk²₂− |x^Hr|²), if Aj =A^H_j λ^j

kΛmk²₂F+ λ^jr^HxPxrr^HPx

kΛmk²₂(krk²₂− |x^Hr|²), if Aj =−A^H_j . and forλ∈C\R,define

4Aj:=











e^T_jbrxx^H+ 1

kΛmk²₂[λ^jPxrx^H+λ^jxr^HPx]−e^T_jr Pb xrr^HPx

krk²₂− |x^Hr|², ifAj=A^H_j ie^T_jrxxb ^H+ 1

kΛmk²₂[λ^jPxrx^H−λ^jxr^HPx] +ie^T_jbr Pxrr^HPx

krk²₂− |x^Hr|², ifAj=−A^H_j . Consider the polynomial 4P(z) = P_m

j=0z^j4Aj. Then 4P ∈ S,P(λ)x+4P(λ)x = 0 and

|||4P|||2=η^S₂(λ, x,P).

Proof: Suppose P ∈ Pm(C^n×n) is an H-Hermitian matrix polynomial. By Theorem 4.2.1 there exists an H-Hermitian matrix polynomial 4P(z) =P_m

j=0z^j4Aj such that 4P(λ)x+ P(λ)x= 0.Now choosing a unitary matrixQ:= [x, Q1], we have

4A]j :=Q^H4AjQ=

à ajj a^H_j aj Xj

!

, Q^Hr=

à x^Hr Q^H₁ r

! .

Now 4P(λ)x+ P(λ)x= 0 ⇒

à P_m

j=0λ^jajj

P_m

j=0λ^jaj

!

=

à x^Hr Q^H₁ r

!

.The minimum norm solution ofP_m

j=0λ^jaj=Q^H₁r is given byaj =_kΛ^λj

mk²₂Q^H₁r.

If λ ∈ R then minimum norm solution of P_m

j=0λ^jajj = x^Hr is ajj = _kΛ^λj

mk²₂x^Hr ∈ R.

Hence for λ∈Rwe have 4Aj=Q

à _λj

kΛmk²₂x^Hr (_kΛ^λj

mk²₂Q^H₁r)^H

λ^j

kΛmk²₂Q^H₁ r Xj

!

Q^H, j= 0 :m. (4.5)

Setting Xj = 0 we obtainη_F^S(λ, x,P) =

√2krk²₂−|r^Hx|²

kΛmk2 .Simplifying the expression of4Aj we obtain the desired result.

Ifλ∈C\R, thenP_m

j=0λ^jajj =x^Hrgives à P_m

j=0re(λ^j)ajj

P_m

j=0im(λ^j)ajj

!

=

à re(x^Hr) im(x^Hr)

!

⇒





 a00

... amm





=M^†

à re(x^Hr) im(x^Hr)

!

=:br(say)

where M =

à Re(Λm)^T Im(Λm)^T

!

. Therefore ajj = e^T_jbr, where ej is the j-th column of identity

matrix. SettingXj= 0 we obtain

η_F^S(λ, x,P) = s

kbrk²₂+ 2krk²₂− |r^Hx|² kΛmk²₂ . Simplifying the expressions of4Ai we obtain the desired result.

Next consider the spectral norm. Letλ∈R.For µ4Aj = ^|λ_kΛ^{j| krk2}

mk²₂ , j = 0 :m, by DKW Theorem 1.2.5 and (4.5) we obtain

Xj=−λ^j x^Hr(Q^H₁ r)(Q^H₁ r)^H kΛmk²₂ (krk²₂− |x^Hr|²). This gives,η^S(λ, x,P) =_kΛ^krk2

mk2 =η(λ, x,P).Simplifying the expression of4Ai we obtain the desired result.

Now if λ ∈ C\R, then for µ4A_j = q

|e^T_jbr|²+^|λj|2(krk_kΛ^22−|xHr|2)

mk⁴₂ applying the DKW Theorem 1.2.5 to 4Aj we have

Xj=−e^T_jbr(Q^H₁r)(Q^H₁ r)^H

krk²₂− |x^Hr|² , j= 0 :m.

This gives,

η₂^S(λ, x,P) = s

kbrk²₂+krk²₂− |x^Hr|² kΛmk²₂ . Simplifying the expression of4A_j, j= 0 :mwe obtain

4Aj =e^T_jbrxx^H+ 1

kΛmk²₂[λ^jPxrx^H+λ^jxr^HPx]−e^T_jr Pb xrr^HPx

krk²₂− |x^Hr|². The proof is similar when P isH-skew-Hermitian polynomial.¥

Remark 4.3.10. If|x^Hr|=krk2,thenkQ^H₁rk2= 0. In such a case, consideringXj= 0, j = 0 :m, we obtain the desired results.

Theorem 4.3.11. Let S be the space ofH-even matrix polynomials. LetP∈S be given by P(z) =P_m

j=0z^jAj.Then for (λ, x)∈C×Cⁿ,setting r=−P(λ)x,we have

η_F^S(λ, x,P) =







√2krk²₂−|x^Hr|² kΛmk2 ≤√

2η(λ, x,P), if λ∈iR q

kbrk²₂+^2(krk_kΛ^22−|xHr|2)

mk²₂ , if λ∈C\iR.

η^S₂(λ, x,P) =







η(λ, x,P), ifλ∈iR q

kbrk²₂+^krk_kΛ^22−|xHr|2

mk²₂ , ifλ∈C\iR where br=

"

ΠeRe(Λm)^T −(I−Πe)Im(Λm)^T ΠeIm(Λm)^T + (I−Πe)Re(Λm)^T

#_†"

re(x^Hr) im(x^Hr)

# . Set

Ej:= 1

kΛmk²₂[λ^jPxrx^H+λ^jxr^HPx], Fj:= 1

kΛmk²₂[λ^jPxrx^H−λ^jxr^HPx]. (4.6)

Whenλ∈iR, define

4A_j := λ^j

kΛmk²₂[xr^H+rx^H−(r^Hx)xx^H].

Forλ∈C\iR,define

4Aj :=

( e^T_jbrxx^H+E_j, ifj is even ie^T_jbrxx^H+Fj, ifj is odd.

Then4P(z) =P_m

j=0z^j4Aj.is a unique polynomial such that4P∈S,P(λ)x+4P(λ)x= 0 and|||4P|||F =η_F^S(λ, x,P).

Further, when λ∈iRdefine 4Aj:= λ^j

kΛmk²₂[xr^H+rx^H−(r^Hx)xx^H] +(−1)^j+1λ^j x^HrPxrr^HPx

kΛmk²₂ (krk²₂− |x^Hr|²) and forλ∈C\iR,define

4Aj:=











e^T_jrxxb ^H+E_j+(−1)^j+1e^T_jbr Pxrr^HPx

krk²₂− |x^Hr|² , if j is even ie^T_jrxxb ^H+Fj+(−1)^j+1ie^T_jbr Pxrr^HPx

krk²₂− |x^Hr|² , if j is odd.

Then 4P(z) = P_m

j=0z^j4Aj is such that 4P ∈ S,P(λ)x+4P(λ)x = 0 and |||4P|||2 = η^S₂(λ, x,P).

Proof: By Theorem 4.2.1, there exists a H-even matrix polynomial4P(z) =P_m

j=0z^j4Aj

such that 4P(λ) =r. Now choosing a unitary matrix Q:= [x, Q1], and noting that4Aj = 4A^H_j whenjis even, and4Aj=−4A^H_j whenjis odd, we have4Aj =Q

Ã

ajj a^H_j aj Xj

! Q^H, X_j^H =Xj ifj is even, and4Aj =Q

à iajj a^H_j

−aj Yj

!

Q^H, Y_j^H =−Yj ifj is odd. Notice that ajj is real for allj.

Then 4P(λ)x=r gives à P

j-evenλ^jajj +iP

j-oddλ^jajj

P

j-evenλ^jaj−P

j-oddλ^jaj

!

=

à x^Hr Q^H₁ r

!

. The mini- mum norm solution ofP

j-evenλ^jaj−P

j-oddλ^jaj =Q^H₁ris given byaj = _kΛ^λj

mk²₂Q^H₁r,ifj is even and aj =−_kΛ^λj

mk²₂Q^H₁r, ifj is odd.

Ifλ∈i R,then the minimum norm solution forajj are given byajj = _kΛ^λj

mk²₂x^Hrwhenj is even, andajj =−_kΛ^{i λj}

mk²₂x^Hrwhenj is odd. Thenajj ∈R,whenj is even andiajj ∈iRif j is odd. Hence if λ∈iR,then we have

4Aj =Q







λ^j

kΛmk²₂x^Hr (_kΛ^λj

mk²₂Q^H₁r)^H

λ^j

kΛmk²₂Q^H₁r Xj





Q^H

whenj is even, and

4Aj=Q







λ^j

kΛmk²₂x^Hr −(_kΛ^λj

mk²₂Q^H₁r)^H

λ^j

kΛmk²₂Q^H₁ r Yj





Q^H

whenj is odd. SettingXj= 0 =Yj,we obtainη_F^S(λ, x,P) =

√2krk²₂−|r^Hx|²

kΛmk2 .Now simplifying the expressions for4Aj we obtain the desired result.

Next ifλ∈C\iR,then P

j-evenλ^jajj+iP

j-oddλ^jajj =x^Hr gives à P

j-evenre(λ^j)ajj−P

j-oddim(λ^j)ajj

P

j-evenim(λ^j)ajj+P

j-oddre(λ^j)ajj

!

=

à re(x^Hr) im(x^Hr)

! .

Hence we have







 a00

a11

... amm







 = K^† Ã

re(x^Hr) im(x^Hr)

!

=:br(say)⇒ajj =e^T_jrb

where K=

à ΠeRe(Λm)^T −(I−Πe)Im(Λm)^T ΠeIm(Λm)^T + (I−Πe)Re(Λm)^T

!

.Consequently we have

η_F^S(λ, x,P) = s

kbrk²₂+ 2krk²₂− |x^Hr|² kΛmk²₂ .

Simplifying the expression of4Aj, j= 0 :mwe obtain the desired result.

Now we consider spectral norm. Forλ∈iR,consider µ4A_j = ^|λ_kΛ^{j| krk2}

mk²₂ ifj is even, and µ_4Aj = ^|λ_kΛ^{j| krk2}

mk²₂ whenj is odd. Then by DKW Theorem 1.2.5 applied to4A_j,gives Xj=−λ^j x^Hr(Q^H₁ r)(Q^H₁ r)^H

kΛmk²₂ (krk²₂− |x^Hr|²), Yj = λ^j x^Hr(Q^H₁r)(Q^H₁ r)^H kΛmk²₂ (krk²₂− |x^Hr|²). This gives η^S₂(λ, x,P) = _kΛ^krk2

mk2. Simplifying the expressions for 4Aj we obtain the desired result.

For λ∈ C\iR, consider µ4Aj = q

|e^T_jbr|²+^|λj|2(krk_kΛ^22−|xHr|2)

mk⁴₂ , for j = 0 :m. Then by DKW Theorem 1.2.5 applied to4Aj gives

Xj =−e^T_jrb(Q^H₁r)(Q^H₁r)^H

krk²₂− |x^Hr|² , Yj= ie^T_jrb(Q^H₁r)(Q^H₁r)^H krk²₂− |x^Hr|² . This gives,

η₂^S(λ, x,P) = s

kbrk²₂+krk²₂− |x^Hr|² kΛmk²₂ .

Simplifying the expression of4Aj, j= 0 :mwe obtain the desired result.¥

Theorem 4.3.12. Let S be the space of H-odd matrix polynomials. Let P∈ S be given by P(z) =P_m

j=0z^jAj.Then for (λ, x)∈C×Cⁿ,setting r=−P(λ)x,we have

η_F^S(λ, x,P) =







√2krk²₂−|x^Hr|² kΛmk2 ≤√

2η(λ, x,P), if λ∈iR q

kbrk²₂+^2(krk_kΛ^22−|xHr|2)

mk²₂ , if λ∈C\iR.

η^S₂(λ, x,P) =







η(λ, x,P), ifλ∈iR q

kbrk²₂+^krk_kΛ^22−|xHr|2

mk²₂ , ifλ∈C\iR where br=

"

−ΠeIm(Λm)^T+ (I−Πe)Re(Λm)^T ΠeRe(Λm)^T + (I−Πe)Im(Λm)^T

#_†"

re(x^Hr) im(x^Hr)

# . Whenλ∈iRdefine

4A_j := λ^j

kΛmk²₂[xr^H−rx^H+ (r^Hx)xx^H].

Forλ∈C\iR, define

4Aj :=

( ie^T_jbrxx^H+F_j, ifj is even e^T_jbrxx^H+Ej, ifj is odd whereEj andFj are given in (4.6). Then4P(z) =P_m

j=0z^j4Aj is a unique polynomial such that 4P∈S,P(λ)x+4P(λ)x= 0and |||4P|||F =η_F^S(λ, x,P).

Further, when λ∈iRdefine 4Aj:= λ^j

kΛmk²₂[xr^H−rx^H+ (r^Hx)xx^H] + (−1)^jλ^j x^HrPxrr^HPx

kΛmk²₂ (krk²₂− |x^Hr|²). When λ∈C\iR,define

4Aj :=















ie^T_jbrxx^H+Fj+(−1)^jie^T_jr Pb xrr^HPx

krk²₂− |x^Hr|² , ifj is even

e^T_jbrxx^H+Ej+(−1)^je^T_jr Pb _xrr^HP_x

krk²₂− |x^Hr|² , ifj is odd.

Then 4P(z) = P_m

j=0z^j4Aj is such that 4P ∈ S,P(λ)x+4P(λ)x = 0 and |||4P|||2 = η^S₂(λ, x,P).

Proof: The proof is similar to Theorem 4.3.11.¥

We mention that the results obtained above can be extended easily to polynomials having more general structures such as the case when the coefficient matrices are in Jordan and/or Lie algebras. Indeed, letM be a unitary matrix such thatM^T =M orM^T =−M.Consider the Jordan algebra J := {A ∈ C^n×n : M⁻¹A^TM = A} and the Lie algebra L := {A ∈ C^n×n : M⁻¹A^TM = −A} associated with the scalar product (x, y) 7→ y^TM x. Consider a polynomial P(z) =P_m

j=0z^jAj, where Aj’s are in J and/or in L. Then the polynomialMP

given byMP(z) =P_m

j=0λ^jM Ajis eitherT-symmetric,T-skew-symmetric,T-even orT-odd.

Hence replacing Aj and r by M Aj and M r, respectively, in the above results, we obtain corresponding results for the polynomial P.

Similarly, whenM is unitary andM =M^HorM =−M^H,we consider the Jordan algebra J := {A ∈ C^n×n : M⁻¹A^HM = A} and the Lie algebra L := {A ∈ C^n×n : M⁻¹A^HM =

−A} associated with the scalar product (x, y) 7→y^HM x.Now, let P(z) = P_m

j=0z^jAj be a polynomial where Aj’s are in J and/or in L. Then the polynomial MP(z) = P_m

j=0z^jM Aj

is either H-Hermitian, H-skew-Hermitian, H-even or H-odd. Hence the results obtained above extend easily to the polynomial P by replacing Aj, r by M Aj, M r, respectively. In particular, whenM :=J,whereJ :=

Ã

0 I

−I 0

!

∈C^2n×2n,the Jordan algebraJconsists of skew-Hamiltonian matrices and the Lie algebra L consists of Hamiltonian matrices. So, for example, considering the polynomial P(z) :=P_m

j=0z^jAj,whereAjs are Hamiltonian whenj is even and skew-Hamiltonian whenjis odd, we see that the polynomialJP(z) =P_m

j=0z^jJA_j is H-even. Hence extending the results obtained for H-even polynomial to the case of P,we have the following.

Theorem 4.3.13. LetSbe the space of polynomials of the formP(z) =P_m

j=0z^jAj whereAj

is Hamiltonian when j is even, andAj is skew-Hamiltonian whenj is odd. Let P∈S. Then for(λ, x)∈C×Cⁿ, setr:=−P(λ)x.Then we have

η_F^S(λ, x,P) =







√2krk²₂−|x^HJr|² kΛmk2 ≤√

2η(λ, x,P), if λ∈iR q

kbrk²₂+^2(krk_kΛ^{22−|xHJr|2)}

mk²₂ , if λ∈C\iR.

η₂^S(λ, x,P) =







η(λ, x,P), ifλ∈iR q

kbrk²₂+^krk22_kΛ^−|xHJr|2

mk²₂ , ifλ∈C\iR where br=

"

Πe(Re(Λm)^T)−(I−Πe)(Im(Λm)^T) Π_e(Im(Λ_m)^T) + (I−Π_e)(Re(Λ_m)^T)

#_†"

re(x^HJr) im(x^HJr)

# .

4.4 Structured backward error and structured lineariza-