Spectral norm structured backward error

∆W_k(z) := ∆A_k+ _j=1z_j∆B_kj fork = 1,· · · , mand∆W(z) := (∆W₁(z),· · · ,∆W_m(z)).

Then ∆W is T-symmetric, W(λ)x+ ∆W(λ)x= 0 and |||∆W|||^p =η_p^S(λ, x,W).

Proof. Recall from the proof of Theorem 6.3.1 that

∆A_k = Q_k





y0

||(1,λ)||^qx^T_krk y0

||(1,λ)||^q(Q^(k)Trk)^T

y0

||(1,λ)||qQ^(k)Trk Ack



Q^H_k,

∆Bkj = Qk





yj

||(1,λ)||^qx^T_kr_{k ||(1,λ)}^yj_||

q(Q^(k)Tr_k)^T

yj

||(1,λ)||^q dBkj



Q^H_k .

satisfies thatWk(λ)xk+ ∆Wk(λ)xk = 0. When||rk||² 6=|x^T_krk|, forµ∆Ak := _||(1,λ)^|y0|_||

q||rk||² and µ_∆Bkj := _||(1,λ)^|yj|_||

q||r_k||2, by Theorem 6.4.1, we have Ack=−y0 x^T_krk(Q^(k)Trk)(Q^(k)Trk)^T

||(1, λ)||^q(||rk||²2− |x^T_krk|²) and dBkj =−yj x^T_krk(Q^(k)Trk)(Q^(k)Trk)^T

||(1, λ)||^q(||rk||²2− |x^T_krk|²). (6.40) When ||rk||² = |x^T_krk| for some k ∈ {1,· · ·, m} then note that ||Q^(k)Trk||² = (||rk||²2 −

|x^T_kr_k|²)¹² = 0, i.e., Q^(k)Tr_k= 0. So we have

∆Ak=Qk





y0

||(1,λ)||^qx^T_krk 0 0 Ack



Q^H_k, ∆Bkj =Qk





yj

||(1,λ)||^qx^T_krk 0 0 Bdkj



Q^H_k.

For µ∆Ak := _||(1,λ)^|y0|_||

q|x^T_krk|= _||(1,λ)^|y0|_||

q||rk||² and µ∆Bkj := _||(1,λ)^|yj|_||

q||rk||², by Theorem 6.4.1, we have Ack = dBkj = 0. So ||∆Wk||^p = _||(1,λ)^||y||p_||

q||rk||² = _||(1,λ)¹ _||

q||rk||². Therefore η_p^S(λ, x,W) =|||∆W|||^p = _||(1,λ)¹ _||

q ||(||r1||²,· · · ,||rm||²)||^p.

When||rk||² 6=|x^T_krk|, substitutingAck andBdkj as given in (6.40) in (6.14) and (6.15), respectively, and when ||rk||² = |x^T_krk|, setting Ack = Bdkj = 0, in (6.14) and (6.15) we obtain the desired perturbation.

Next we consider T-skew symmetric MEPs.

Theorem 6.4.3. Let S be the space of all T-skew symmetric MEPs and W ∈ S be as given in (6.1). Let λ := (λ1,· · ·, λm)^T ∈ C^m and 0 6= x := x1 ⊗ · · · ⊗xm ∈ Cⁿ¹⊗ · · · ⊗C^nm be such thatx^H_kxk = 1. Consider rk:=−Wk(λ)xk for all k = 1,· · · , m.

Then η^S_p(λ, x,W) = _||(1,λ)¹ _||

q ||(||r1||²,· · · ,||rm||²)||^p, where p⁻¹+q⁻¹ = 1.

Let (y₀,· · · , y_m) ∈ ∂||(1, λ)||q. Define ∆A_k = _||(1,λ)||

q (r_kx_k −x_kr_k) and ∆B_kj =

yj

||(1,λ)||^q (rkx^H_k −xkr_k^T)for all k, j = 1,· · · , m. Consider∆Wk(z) := ∆Ak+Pm

j=1zj∆Bkj

for k = 1,· · · , m and ∆W(z) := (∆W₁(z),· · · ,∆W_m(z)). Then ∆W is T-skew sym- metric, W(λ)x+ ∆W(λ)x= 0 and |||∆W|||^p =η_p^S(λ, x,W).

Proof. Recall from the proof of Theorem 6.3.2 that in this case we have

∆Ak = Qk



 0 −_||(1,λ)^y0 _||q(Q^(k)Trk)^T

y0

||(1,λ)||qQ^(k)Trk Ack



Q^H_k,

∆Bkj = Qk



 0 −_||(1,λ)^yj_||q(Q^(k)Tr_k)^T

yj

||(1,λ)||^qQ^(k)Trk dBkj



Q^H_k.

such that Wk(λ)xk + ∆Wk(λ)xk = 0. Now for µ∆Ak := _||(1,λ)^|y0|_||

q||rk||² and µ∆Bkj :=

|yj|

||(1,λ)||^q||rk||²by Theorem 6.4.1, we haveAck =Bdkj = 0. Thenη_p^S(λ, xk, Wk) = _||(1,λ)^||y||p_||

q||rk||² =

1

||(1,λ)||q||rk||² and thus we have η_p^S(λ, x,W) = _||(1,λ)¹ _||

q ||(||r1||²,· · · ,||rm||²)||^p. Further simplifying ∆Ak and ∆Bkj, we have

∆Ak= y0

||(1, λ)||^q(rkx^H_k −xkr_k^T)+Q^(k)AckQ^(k)H,∆Bkj = yj

||(1, λ)||^q(rkx^H_k −xkr^T_k)+Q^(k)BdkjQ^(k)H. Setting Ac_k=dB_kj = 0, we obtain the desired optimal perturbation.

Now we derive η^S_p(λ, x,W) for a T-even MEP.

Theorem 6.4.4. Let S be the space of all T-even MEPs and W ∈ S be as given in (6.1). Let λ := (λ1,· · · , λm)^T ∈ C^m and 0 6= x := x1 ⊗ · · · ⊗xm ∈ Cⁿ¹ ⊗ · · · ⊗ C^nm be such that x^H_kxk = 1. Consider rk := −Wk(λ)xk for all k = 1,· · · , m. Let (y₀,· · · , y_m)^T ∈ ∂||(1, λ)||q, for p⁻¹ +q⁻¹ = 1. Then η_p^S(λ, x,W) = ||(c₁,· · · , c_m)||p, where ck =||(ak, bk1,· · · , bkm)||^p, ak =

|x^T_kAkxk|²+ _||(1,λ)^|y0|2_||2 q

||rk||²2− |x^T_kAkxk|²¹2

and bkj = _||(1,λ)^|yj|_||q ||rk||²2− |x^T_kAkxk|²¹₂

for all k, j = 1,· · ·, m. In particular, when p = 2, we have η₂^S(λ, x,W) =Pm

k=1

|x^T_kAkxk|²+_||(1,λ)¹ _||2

2 ||rk||²2− |x^T_kAkxk|²¹2

.

When ||r_k||2 6=|x_kA_kx_k|, define

∆Ak := −xkx^T_kAkxkx^H_k + y0

||(1, λ)||^q

rkx^H_k +xkr_k^T −2(x^T_kAkxk)xkx^H_k

−y02 x^T_kAkxk(Ink−xkx^T_k)rkr_k^T(Ink −xkx^H_k)

|y0|²(||rk||²2− |x^T_kAkxk|²) ,

∆Bkj := yj

||(1, λ)||^q rkx^H_k −xkr^T_k and when ||r_k||2 =|x^T_kA_kx_k|, define

∆Ak := −xkx^T_kAkxkx^H_k + y0

||(1, λ)||^q

rkx^H_k +xkr_k^T −2(x^T_kAkxk)xkx^H_k ,

∆Bkj := y_j

||(1, λ)||^q rkx^H_k −xkr^T_k

for all k, j = 1,· · · , m. Consider∆Wk(z) := ∆Ak+Pm

j=1zj∆Bkj fork = 1,· · · , m and

∆W(z) := (∆W1(z),· · · ,∆Wm(z)). Then ∆W is T-even, W(λ)x+ ∆W(λ)x = 0 and

|||∆W|||^p =η_p^S(λ, x,W).

Proof. It follows from the proof of Theorem 6.3.3 that we have

∆Ak = Qk



 −x^T_kAkxk y0

||(1,λ)||^q(Q^(k)Trk)^T

y0

||(1,λ)||^qQ^(k)Trk Ack



Q^H_k,

∆Bkj = Qk



 0 −_||(1,λ)^yj_||q(Q^(k)Tr_k)^T

yj

||(1,λ)||^qQ^(k)Trk dBkj



Q^H_k

such that Wk(λ)xk+ ∆Wk(λ)xk = 0. When ||rk||² 6=|x^T_kAkxk| for some k∈ {1,· · · , m} for µ∆Ak =ak and µ∆Bkj =bkj, by Theorem 6.4.1, we have

Ack =−y₀² x^T_kA_kx_k(Q^(k)Tr_k)(Q^(k)Tr_k)^T

|y0|²(||rk||²2− |x^T_kAkxk|²)¹² and dBkj = 0. (6.41) When ||rk||² = |x^T_kAkxk| for some k ∈ {1,· · · , m}, note that Q^(k)Trk = 0 and thus

∆A_k = Q_k



 −x^T_kAkxk 0 0 Ack



Q^H_k and ∆B_kj = Q_k



 0 0 0 dBkj



Q^H_k. Again for µ_∆Ak = ak and µ∆Bkj = bkj, by Theorem 6.4.1, we have Ack = dBkj = 0. Then ||∆Wk||² =

||(a_k, b_k1,· · · , b_km)||p =c_k and therefore η_p(λ, x,W) = |||∆W|||p = ||(c₁,· · · , c_m)||p. Fur- ther simplifying ∆Ak and ∆Bkj, we have

∆Ak = −xkx^T_kAkxkx^H_k + y0

||(1, λ)||q

rkx^H_k +xkr_k^T −2(x^T_kAkxk)xkx^H_k

+Q^(k)AckQ^(k)H

∆Bkj = yj

||(1, λ)||^q rkx^H_k −xkr^T_k

+Q^(k)BdkjQ^(k)H.

When ||rk||² 6=|x^T_kAkxk| for somek ∈ {1,· · · , m}, settingAckand Bdkj as given in (6.41) and when||rk||² =|x^T_kAkxk|for somek ∈ {1,· · ·, m}, settingAck=dBkj = 0, the desired perturbation follows.

In case of a T-odd MEP, we have the following result.

Theorem 6.4.5. Let Sbe the space of all T-odd MEPs and W∈S be as given in (6.1).

Let λ := (λ1,· · · , λm)^T ∈ C^m and 0 6= x := x1 ⊗ · · · ⊗xm ∈ Cⁿ¹ ⊗ · · · ⊗C^nm be such that x^H_kxk = 1. Consider rk := −Wk(λ)xk for all k = 1,· · · , m. Let (y0,· · · , ym)^T ∈

∂||(1, λ)||^q and (z1,· · · , zm)^T ∈∂||λ||^q, for p⁻¹+q⁻¹ = 1. Then

η^S_p(λ, x,W) =





||(c1,· · · , cm)||^p when λ6= 0

||(||r1||²,· · · ,||rm||²)||^p when λ= 0, where ck =||(ak, bk1,· · · , bkm)||^p and

ak = |y0|

||(1, λ)||^q(||rk||²2−|x^T_krk|²)¹² andbkj =

|zj|²

||λ||²q|x^T_krk|²+ |yj|²

||(1, λ)||²q

(||rk||²2− |x^T_krk|²) ¹₂

for all k, j = 1,· · · , m. In particular, when p= 2, we have

η^S₂(λ, x,W) =



 Pm

k=1

1

||λ||²2|x^T_krk|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^T_kAkxk|²¹2

when λ6= 0

||(||r1||²,· · · ,||rm||²)||² when λ= 0.

When ||rk||² 6=|x^T_krk| for some k∈ {1,· · · , m}, define

∆Ak := y0

||(1, λ)||q

(rkx^H_k −xkr_k^T)

∆Bkj := zj

||λ||^q(x^T_krk)xkx^H_k + yj

||(1, λ)||^q(rkx^H_k +xkr_k^T −2(x^T_krk)xkx^H_k)

−yj2 zj x^T_krk(Ink−xkx^T_k)rkr_k^T(Ink−xkx^H_k)

||λ||^q |yj|² (||rk||²2− |x^T_krk|²)

and when ||r_k||2 =|x_kr_k| for some k ∈ {1,· · · , m}, define

∆Ak := y0

||(1, λ)||^q(rkx^H_k −xkr^T_k),

∆Bkj := zj

||λ||^q(x^T_krk)xkx^H_k + yj

||(1, λ)||^q(rkx^H_k +xkr_k^T −2(x^T_krk)xkx^H_k) for all k, j = 1,· · · , m. Consider∆Wk(z) := ∆Ak+Pm

j=1zj∆Bkj fork = 1,· · · , m and

∆W(z) := (∆W1(z),· · · ,∆Wm(z)). Then ∆W is T-odd, W(λ)x+ ∆W(λ)x = 0 and

|||∆W|||^p =η_p^S(λ, x,W).

Now we consider T-even alternating MEPs.

Theorem 6.4.6. Let S be the space of all T-even alternating MEPs and W ∈ S be as given in (6.1). Let λ := (λ1,· · ·, λm)^T ∈ C^m and 0 6= x := x1 ⊗ · · · ⊗xm ∈ Cⁿ¹⊗ · · · ⊗C^nm be such thatx^H_kxk = 1. Consider rk:=−Wk(λ)xk for all k = 1,· · · , m.

Let (y0,· · · , ym)^T ∈ ∂||(1, λ)||^q and (z0,· · ·, zm)^T ∈ ∂||(1,Π(e)(λ)||^q for p⁻¹ +q⁻¹ = 1.

Then η^S_p(λ, x,W) =||(c1,· · · , cm)||^p, where ck =||(ak, bk1,· · · , bkm)||^p and ak =

|z0|²

||(1,Λ^(e)(λ))||²q|x^T_krk|²+ |y0|²

||(1, λ)||²q

(||rk||²2− |x^T_krk|²) ¹₂

bkj =







|zj|²

||(1,Λ^(e)(λ))||²q|x^T_krk|²+ _||(1,λ)^|yj|2_||2

q(||rk||²2− |x^T_krk|²)¹₂

when j is even

|yj|

||(1,λ)||^q(||rk||²2− |x^T_krk|²)¹² when j is odd for all k, j = 1,· · · , m. In particular, for p= 2, we have

η^S₂(λ, x,W) = Xm k=1

1

||(1,Λ^(e)(λ))||²2|x^T_krk|²+ 1

||(1, λ)||²2 ||rk||²2− |x^T_krk|²!¹² . When ||rk||² 6=|x^T_krk| for some k ∈ {1,· · · , m}, define

∆Ak := z0

||(1,Λ^(e)(λ))||q

(x^T_krk)xkx^H_k + y0

||(1, λ)||q

[rkx^H_k +xkr^T_k −2(x^T_krk)xkx^H_k]

−y₀²z₀x^T_kr_k(I_nk−x_kx^T_k)r_kr_k^T(I_nk −x_kx^H_k)

|y0|²(||rk||²2− |x^T_krk|²)||(1,Λ^(e)(λ))||^q ,

∆Bkj :=













zj

||(1,Λ^(e)(λ))||^q(x^T_krk)xkx^H_k +_||(1,λ)^yj _||

q[rkx^H_k +xkr_k^T −2(x^T_krk)xkx^H_k]

−_|yj|²_(||rk||²_2−|^y_x^j2TzjxTkrk

krk|²)||(1,Λ^(e)(λ))||^q(Ink −xkx^T_k)rkr^T_k(Ink −xkx^H_k ) when j is even

yj

||(1,λ)||^q[rkx^H_k −xkr^T_k] when j is odd

and when ||r_k||2=|x_kr_k| for some k ∈ {1,· · · , m}, define

∆Ak := z0

||(1,Λ^(e)(λ))||^q(x^T_krk)xkx^H_k + y0

||(1, λ)||^q[rkx^H_k +xkr_k^T −2(x^T_krk)xkx^H_k],

∆Bkj :=







zj

||(1,Λ^(e)(λ))||q(x^T_krk)xkx^H_k +_||(1,λ)^yj_||

q[rkx^H_k +xkr_k^T −2(x^T_krk)xkx^H_k] when j is even

yj

||(1,λ)||^q[rkx^H_k −xkr_k^T] when j is odd

for all k, j = 1,· · · , m. Consider ∆Wk(z) := ∆Ak +Pm

j=1zj∆Bkj for k = 1,· · ·, m and ∆W(z) := (∆W1(z),· · · ,∆Wm(z)). Then ∆W is T-even alternating, W(λ)x+

∆W(λ)x= 0 and |||∆W|||^p =η_p^S(λ, x,W).

Proof. Recall from the proof of Theorem 6.3.5 that

∆Ak = Qk





z0

||(1,Π_(e)(λ))||^qx^T_krk y0

||(1,λ)||^q(Q^(k)Trk)^T

y0

||(1,λ)||^qQ^(k)Trk Ack



Q^H_k,

∆Bkj =



















 Qk





zj

||(1,Π_(e)(λ))||qx^T_krk yj

||(1,λ)||q(Q^(k)Trk)^T

yj

||(1,λ)||qQ^(k)Trk Bdkj



Q^H_k when j is even

Qk



 0 −_||(1,λ)^yj _||q(Q^(k)Trk)^T

yj

||(1,λ)||^qQ^(k)Trk Bdkj



Q^H_k when j is odd

for all k, j = 1,· · · , m. When ||rk||² 6=|x^T_krk| for some k ∈ {1,· · · , m}, for µ∆Ak = ak

and µ∆Bkj =bkj, by Theorem 6.4.1, we have Ack =− y02z0x^T_krk

|y0|²(||rk||²2− |x^T_krk|²)||(1,Π(e)(λ))||^q(Q^(k)Trk)(Q^(k)Trk)^T, dBkj =





−_|yj|2(||rk||²2−|^yx^j2T_k^zr^jk^x|^Tk2)^r||^k(1,Π_(e)(λ))||^q(Q^(k)Tr_k)(Q^(k)Tr_k)^T when j is even

0 when j is odd.

When ||rk||² = |x^T_krk| for some k ∈ {1,· · · , m} then noting that Q^(k)Trk = 0 we have

∆Ak = Qk





z0

||(1,Π(e)(λ))||^qx^T_krk 0

0 Ack



Q^H_k , ∆Bkj =Qk





zj

||λ||^qx^T_krk 0 0 Bd_kj



Q^H_k when j is

even and ∆B_kj =Q_k



 0 0 0 Bdkj



Q^H_k when j is odd. Again forµ_∆Ak =a_k and µ_∆Bkj =

bkj, by Theorem 6.4.1, we have Ack =Bdkj = 0. So ||∆Wk||^p =||(ak, bk1,· · · , bkm)||^p =ck

and hence η_p^S(λ, x,W) = |||∆W|||^p = ||(c1,· · · , cm)||^p. Simplifying ∆Ak and ∆Bkj, the desired perturbation follows.

Similarly we have the following result for aT-odd alternating MEP.

Theorem 6.4.7. Let S be the space of all T-odd alternating MEPs and W ∈ S be as given in (6.1). Let λ := (λ1,· · ·, λm)^T ∈ C^m and 0 6= x := x1 ⊗ · · · ⊗xm ∈ Cⁿ¹⊗· · ·⊗C^nm be such that x^H_kxk = 1. Considerrk :=−Wk(λ)xk, for allk = 1,· · · , m.

Let(y0,· · · , ym)^T ∈∂||(1, λ)||^qand(z1,· · · , zm)^T ∈∂||Π(o)(λ)||^q forp⁻¹+q⁻¹ = 1. Then η_p^S(λ, x,W) =





||(c1,· · · , cm)||^p when Π(o)(λ)6= 0

1

||(1,λ)||^q||(||r1||²,· · · ,||rm||)||^p when Π(o)(λ) = 0, where ck =||(ak, bk1,· · · , bkm)||^p, ak= _||(1,λ)^|y0|_||

q(||rk||²2− |x^T_krk|²)¹² and bkj =







|yj|

||(1,λ)||q(||rk||²2− |x^T_krk|²)¹² when j is even |zj|²

||Λ^(o)(λ)||²q|x^T_kr_k|²+ _||(1,λ)^|yj|2_||2

q(||r_k||²2− |x^T_kr_k|²)¹₂

when j is odd for all k, j = 1,· · · , m. In particular, for p= 2, we have

η₂^S(λ, x,W) =



 Pm

k=1

1

||Λ^(o)(λ))||²2|x^T_krk|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^T_krk|²¹²

when Π(o)(λ)6= 0

1

||(1,λ)||2||(||r1||²,· · · ,||rm||)||² when Π(o)(λ) = 0.

When ||rk||²6=|x^T_krk| for some k ∈ {1,· · · , m}, define∆Ak := _||(1,λ)^y0_||

q[rkx^H_k −xkr_k^T],

∆B_kj :=













yj

||(1,λ)||^q[rkx^H_k −xkr^T_k] when j is even

zj

||Λ^(o)(λ)||^q(x^T_krk)xkx^H_k + _||(1,λ)^yj_||

q[rkx^H_k +xkr^T_k −2(x^T_krk)xkx^H_k]

−_|yj|²_(||rk||²₂^y_−|^j2_x^zTjxTkrk

krk|²)||Λ^(e)(λ)||^q(Ink −xkx^T_k)rkr^T_k(Ink−xkx^H_k) when j is odd and when ||rk||²=|x^T_krk|for some k ∈ {1,· · · , m}, define ∆Ak:= _||(1,λ)^y0 _||

q[rkx^H_k −xkr_k^T],

∆Bkj :=







yj

||(1,λ)||^q[rkx^H_k −xkr^T_k] when j is even

zj

||Λ^(o)(λ)||^q(x^T_kr_k)x_kx^H_k + _||(1,λ)^yj_||

q[r_kx^H_k +x_kr^T_k −2(x^T_kr_k)x_kx^H_k] when j is odd, for all k, j = 1,· · · , m. Consider ∆Wk(z) := ∆Ak+Pm

j=1zj∆Bkj for k = 1,· · · , m and ∆W(z) := (∆W₁(z),· · · ,∆Wm(z)). Then ∆W is T-odd alternating, W(λ)x +

∆W(λ)x= 0 and |||∆W|||^p =η_p^S(λ, x,W).

Now we derive the structured backward error for the H-structures by considering H¨older 2-norm|||·|||² on MEP(n₁,· · · , nm).

Theorem 6.4.8. Let S be the space of all H-Hermitian MEPs and W∈ S be as given in (6.1). Let λ:= (λ₁,· · ·, λ_m)^T ∈C^m be such that λ_j :=α_j+iβ_j for all j = 1,· · · , m, where i=√

−1 and 06=x :=x1⊗ · · · ⊗xm ∈Cⁿ¹ ⊗ · · · ⊗C^nm be such that x^H_kxk = 1.

Consider rk:=−Wk(λ)xk and Pk =Ink−xkx^H_k for all k= 1,· · · , m. Then η^S₂(λ, x,W) =





1

||(1,λ)||²||(||r1||2,· · · ,||rm||2)|| when λ∈R^m Pm

k=1

Pm

j=0|skj|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^H_k rk|²¹2

when λ∈C^m\R^m, where skj is as defined in Theorem 6.3.7.

When ||rk||² 6=|x^H_krk|, for some k∈ {1,· · · , m}, define

∆Ak=







1

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_k xk)xkx^H_k

− _||(1,λ)^xHk_||²₂^r₍^k_||^P_r^k_k^r_||^k2₂^r_−|^kH_x^PHk

krk|²) when λ ∈R^m sk0xkx^H_k +_||(1,λ)¹ _||2

2

Pkrkx^H_k +xkr^H_kPk

−_||^s_r^k0_k||^P2krkrkHPk

2−|x^H_krk|² when λ ∈C^m\R^m,

∆Bkj =







λj

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r_k^Hxk)xkx^H_k

− _||(1,λ)^λjx_||^Hk2₂₍^r_||^k_r^P_k^k_||^r2₂^k_−|^rHk_x^HPk

krk|²) when λ ∈R^m skjxkx^H_k +_||(1,λ)¹ _||2

2

λjPkrkx^H_k +λjxkr^H_kPk

− _||^s_r^kj_k||^P2krkrkHPk

2−|x^H_krk|² when λ ∈C^m\R^m and when ||r_k||26=|x^H_kr_k|, for some k ∈ {1,· · ·, m}, define

∆A_k =





1

||(1,λ)||²2(x^H_k rk)xkx^H_k = _||(1,λ)¹ _||2

2rkx^H_k when λ ∈R^m xksk0x^H_k when λ ∈C^m\R^m,

∆Bkj =





λj

||(1,λ)||²2(x^H_k r_k)x_kx^H_k = _||(1,λ)^λj_||2

2r_kx^H_k when λ ∈R^m xkskjx^H_k when λ ∈C^m\R^m, for all k, j = 1,· · · , m. Consider ∆Wk(z) := ∆Ak+Pm

j=1zj∆Bkj fork = 1,· · · , m and

∆W(z) := (∆W₁(z),· · · ,∆Wm(z)). Then ∆W isH-Hermitian, W(λ)x+ ∆W(λ)x= 0 and |||∆W|||² =η₂^S(λ, x,W).

Proof. When λ∈C^m\R^m, recall from Theorem 6.3.7 that

∆Ak = Qk



 sk0 1

||(1,λ)||2(Q^(k)Hrk)^H

1

||(1,λ)||2Q^(k)Hrk Ack



Q^H_k ,

∆Bkj = Qk



 skj λj

||(1,λ)||2(Q^(k)Hrk)^H

λj

||(1,λ)||²Q^(k)Hrk Bdkj



Q^H_k

for all j = 1,· · ·, m. So forµ∆Ak := |sk0|²+_||(1,λ)¹ _||2

2 ||rk||²2− |x^H_krk|²2

and µ∆Bkj :=

|skj|²+_||(1,λ)^|λj|2_||2

2 ||rk||²2− |x^H_krk|²¹²

by Theorem 6.4.1, we haveAck =−^sk0(Q(k)

Hrk)(Q^(k)Hrk)^H

||rk||²2−|x^H_krk|²

and Bdkj = −^skj(Q(k)

Hrk)(Q^(k)Hrk)^H

||rk||²2−|x^H_krk|² . When ||rk||² = |x^H_krk| for some k ∈ {1,· · · , m} then note that ||Q^(k)Hrk||² = (||rk||²2 − |x^H_krk|²)¹² = 0, i.e., Q^(k)Hrk = 0. So we have

∆Ak = Qk



 sk0 0 0 Ack



Q^H_k and ∆Bkj = Qk



 skj 0 0 Bdkj



Q^H_k . For µ∆Ak = |sk0| and µ∆Bkj = |skj|, again by Theorem 6.4.1, we have Ack = Bdkj = 0. Hence ||∆Wk||² = Pm

j=0|s_kj|²+||r_k||²2− |x^H_kr_k|²¹₂ and

η₂^S(λ, x,W) =|||∆W|||2 = Xm k=1

Xm j=0

|s_kj|²+||r_k||²2− |x^H_kr_k|²

!!¹₂ . When λ∈R^m, recall from Theorem 6.3.7 that

∆A_k = Q_k





1

||(1,λ)||²2x^H_krk 1

||(1,λ)||²2(Q^(k)Hrk)^H

1

||(1,λ)||²2Q^(k)Hrk Abk



Q^H_k,

∆Bkj = Qk





λj

||(1,λ)||²2x^H_krk λj

||(1,λ)||²2(Q^(k)Hrk)^H

λj

||(1,λ)||²2Q^(k)Hrk Bbkj



Q^H_k

for all j = 1,· · · , m. When ||rk||² 6= |x^H_krk| for µ∆Ak = _||(1,λ)¹ _||2

2||rk||² and µ∆Bkj =

|λj|

||(1,λ)||²2||rk||² by Theorem 6.4.1, we have Ack=−x^H_k rk(Q^(k)Hrk)(Q^(k)Hrk)^H

||rk||²2− |x^H_k rk|² and Bdkj =−λj x^H_krk(Q^(k)Hrk)(Q^(k)Hrk)^H

||rk||²2− |x^H_krk|² .

Finally when||rk||² =|x^H_krk|for somek∈ {1,· · · , m}then ∆Ak=Qk





1

||(1,λ)||²2x^H_krk 0 0 Ack



Q^H_k

and ∆Bkj = Qk





λj

||(1,λ)||²2x^H_krk 0 0 dB_kj



Q^H_k. For µ∆Ak = _||(1,λ)¹ _||2

2||rk||² and µ∆Bkj =

|λj|

||(1,λ)||²2||rk||²by Theorem 6.4.1, we haveAck =dBkj = 0. Therefore||∆Wk||² = _||(1,λ)¹ _||2||rk||² and and η₂^S(λ, x,W) = |||∆W|||² = _||(1,λ)¹ _||

2||(||r1||²,· · · ,||rm||²)||². The desired optimal perturbations follow by simplifying ∆Ak and ∆Bkj.

Next we derive η^S₂(λ, x,W) forH-even MEP.

Theorem 6.4.9. Let S be the space of all H-even MEPs and W ∈ S be as given in (6.1). Let λ := (λ1,· · ·, λm)^T ∈ C^m be such that λj := αj +iβj,for all j = 1,· · · , m, where i=√

−1 and 06=x :=x₁⊗ · · · ⊗x_m ∈Cⁿ¹ ⊗ · · · ⊗C^nm be such that x^H_kx_k = 1.

Consider rk:=−Wk(λ)xk and Pk :=Ink−xkx^H_k for all k= 1,· · ·, m. Then

η^S₂(λ, x,W) =





1

||(1,λ)||²||(||r1||²,· · · ,||rm||²)||² when λ∈iR^m Pm

k=1

Pm

j=0|skj|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^H_k rk|²¹2

when λ∈C^m\iR^m where skj are as given in Theorem 6.3.8.

When ||r_k||2 6=|x^H_kr_k|, for some k∈ {1,· · · , m}, define

∆A_k:=





1

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_kxk)xkx^H_k

− _||(1,λ)^x_||^Hk2^rkPkrkrkHPk

2(||rk||²2−|x^H_krk|²) λ∈iR^m s_k0x_kx^H_k + _||(1,λ)¹ _||2

2

P_kr_kx^H_k +x_kr^H_kP_k

−_||^s_r^k0_k||^P2krkrkHPk

2−|x^H_krk|² λ∈C^m\iR^m,

∆Bkj :=





λj

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_kxk)xkx^H_k

+ +_||^is_r^kjPkrkrHkPk

k||²2−|x^H_krk|² λ ∈iR^m iskjxkx^H_k +_||(1,λ)¹ _||2

2

λjPkrkx^H_k −λjxkr_k^HPk

+_||^is_r^kjPkrkrkHPk

k||²2−|x^H_krk|² λ ∈C^m\iR^m When ||r_k||2 =|x^H_kr_k| for some k∈ {1,· · · , m}, define

∆Ak :=





1

||(1,λ)||²2r_kx^H_k λ ∈iR^m

s_k0x_kx^H_k λ ∈C^m\iR^m, ∆Bkj :=





λj

||(1,λ)||²2r_kx^H_k λ∈iR^m is_kjx_kx^H_k λ∈C^m\iR^m for all k, j = 1,· · · , m. Consider ∆W_k(z) := ∆A_k+Pm

j=1z_j∆B_kj fork = 1,· · · , m and

∆W(z) := (∆W₁(z),· · · ,∆Wm(z)). Then ∆W is H-even, W(λ)x+ ∆W(λ)x= 0 and

|||∆W|||² =η₂^S(λ, x,W).

Proof. When λ∈iR^m, recall from the proof of Theorem 6.3.8 that

∆A_k = Q_k





1

||(1,λ)||²2x^H_krk 1

||(1,λ)||²2(Q^(k)Hrk)^H

1

||(1,λ)||²2Q^(k)Hrk Ack



Q^H_k,

∆Bkj = Qk





λj

||(1,λ)||²2x^H_krk −_||(1,λ)^λj_||²₂(Q^(k)Hrk)^H

λj

||(1,λ)||²2Q^(k)Hr_k Bd_kj



Q^H_k

for all j = 1,· · · , m. When ||rk||² 6= |x^H_krk| for µ∆Ak = _||(1,λ)¹ _||2

2||rk||² and µ∆Bkj =

|λj|

||(1,λ)||²2||r_k||2, by Theorem 6.4.1, we have Ac_k = −^xHkrk(Q(k)

Hrk)(Q^(k)Hrk)^H

||(1,λ)||²2(||rk||²2−|x^H_krk|²) and Bd_kj =

j k k k k

||(1,λ)||²2(||rk||²2−|x^H_krk|²) . When ||rk||² = |x^H_krk| for some k ∈ {1,· · · , m}, we have Q^(k)Hrk = 0. Hence

∆Ak =Qk





1

||(1,λ)||²2x^H_krk 0 0 Ack



Q^H_k and ∆Bkj =Qk





λj

||(1,λ)||²2x^H_krk 0 0 Bdkj



Q^H_k.

Again for µ∆Ak = _||(1,λ)¹ _||2

2||rk||² and µ∆Bkj = _||(1,λ)^|λj|_||2

2||rk||² by Theorem 6.4.1, we have Ack = dBkj = 0. So ||∆Wk||² = _||(1,λ)¹ _||2||rk||² and therefore η₂^S(λ, x,W) = |||∆W|||² =

1

||(1,λ)||2||(||r1||²,· · · ,||rm||²)||².

When λ∈C^m\iR^m, recall from Theorem 6.3.8 that

∆Ak = Qk



 s_{k0 ||(1,λ)}¹ _||2

2(Q^(k)Hr_k)^H

1

||(1,λ)||²2Q^(k)Hrk Ack



Q^H_k,

∆Bkj = Qk



 iskj −_||(1,λ)^λj_||²₂(Q^(k)Hrk)^H

λj

||(1,λ)||²2Q^(k)Hrk Bdkj



Q^H_k,

for all j = 1,· · · , m. When ||r_k||2 6= |x^H_kr_k| for some k ∈ {1,· · · , m}, for µ_∆Ak = |sk0|²+ _||(1,λ)¹ _||4

2 ||rk||²2− |x^H_krk|²¹²

andµ∆Bkj =

|skj|²+_||(1,λ)^|λj|2_||4

2 ||rk||²2− |x^H_krk|²¹² by Theorem 6.4.1, we have Ack =−^sk0(Q(k)

Hrk)(Q^(k)Hrk)^H

||rk||²2−|x^H_krk|² and dBkj = ^iskj(Q(k)

Hrk)(Q^(k)Hrk)^H

||rk||²2−|x^H_krk|² . When||r_k||2 =|x^H_kr_k|for somek ∈ {1,· · · , m}then we have ∆A_k =Q_k



 sk0 0 0 Ack



Q^H_k

and ∆Bkj = Qk



 iskj 0 0 Bd_kj



Q^H_k . For µ∆Ak = |sk0| and µ∆Bkj = |skj|, by Theo-

rem 6.4.1, we haveAck =dBkj = 0. Therefore||∆Wk||² =Pm

j=0|skj|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^H_krk|²¹2

and thus η^S₂(λ, x,W) =|||∆W|||² =Pm k=1

Pm

j=0|skj|²+_||(1,λ)¹ _||2

2 ||rk||²2− |x^H_krk|²¹² . SimplifyingAck and Bdkj, we obtain the desired optimal perturbations.

In a similar way, in case of aH-even alternating MEP, we have the following result.

Theorem 6.4.10. Let S be the space of all H-even alternating MEPs and W ∈ S be as given in (6.1). Let λ := (λ₁,· · · , λ_m)^T ∈ C^m be such that λ_j := α_j +iβ_j for all j = 1,· · · , m, where i =√

−1, 06= x:=x1 ⊗ · · · ⊗xm ∈Cⁿ¹ ⊗ · · · ⊗C^nm be such that

x_kx_k = 1. Consider r_k :=−W_k(λ)x_k and P_k :=I_nk−x_kx_k for all k= 1,· · ·, m. Then

η^S₂(λ, x,W) =





1

||(1,λ)||2||(||r₁||2,· · · ,||r_m||2)||2 when λ∈iR^m P_m

k=1

P_m

j=0|skj|²+ _||(1,λ)¹ _||2

2 ||rk||²2− |x^H_k rk|²¹²

when λ∈C^m\iR^m where skj are as given in Theorem 6.3.9.

When ||rk||² 6=|x^H_krk|, for some k∈ {1,· · · , m}, define

∆Ak:=





1

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_kxk)xkx^H_k

− _||(1,λ)^x_||^Hk2^rkPkrkrkHPk

2(||rk||²2−|x^H_krk|²) when λ∈iR^m sk0xkx^H_k + _||(1,λ)¹ _||2

2

Pkrkx^H_k +xkr^H_kPk

−_||^s_r^k0_k||^P2₂^k_−|^rk_x^rHkHPk

krk|² when λ∈C^m\iR^m

∆B_kj :=





λj

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_kxk)xkx^H_k

− _||(1,λ)^λj_||^x2^{HkrkPkrkrHkPk}

2(||rk||²2−|x^H_krk|²) when λ∈iR^m s_k0x_kx^H_k + _||(1,λ)¹ _||2

2

P_kr_kx^H_k +x_kr_k^HP_k

−_||^s_r^kj_k||^P2krkrHkPk

2−|x^H_krk|² when λ∈C^m\iR^m when j is even and

∆Bkj :=





λj

||(1,λ)||²2 rkx^H_k +xkr^H_k −(r^H_kxk)xkx^H_k

+ ^{λjxHkrkPkrkrkHPk}

||(1,λ)||²2(||rk||²2−|x^H_krk|²) when λ∈iR^m iskjxkx^H_k +_||(1,λ)¹ _||2

2

λjPkrkx^H_k −λjxkr^H_k Pk

+_||^is_r^kjPkrkrkHPk

k||²2−|x^H_krk|² when λ∈C^m\iR^m When ||rk||² = |x^H_krk| for some k ∈ {1,· · · , m} define ∆Ak := _||(1,λ)¹ _||2

2rkx^H_k, ∆Bkj :=

λj

||(1,λ)||²2rkx^H_k whenλ ∈iR^m and∆Ak :=sk0xkx^H_k,∆Bkj :=





skjxkx^H_k when j is even is_kjx_kx^H_k when j is odd, when λ ∈ C^m \ iR^m. Consider ∆Wk(z) := ∆Ak + Pm

j=1zj∆Bkj for k = 1,· · ·, m and ∆W(z) := (∆W₁(z),· · · ,∆Wm(z)). Then ∆W is H-even alternating, W(λ)x+

∆W(λ)x= 0 and |||∆W|||² =η₂^S(λ, x,W).

7

Summary

We have developed a general framework for the spectral analysis of nonhomogeneous MEPs. For a general norm on the space of MEPs, we have defined the condition number cond(λ,W) of a simple eigenvalue λ of an MEP Wby

cond(λ,W) := lim sup

|||∆W|||→0

dist(λ, σ(W+ ∆W))

|||∆W||| ,

where ∆W is an MEP and dist(λ, σ(W+ ∆W)) := min{||λ−µ||² : µ∈σ(W+ ∆W)}. Given a simple eigenvalue λW of W, we have shown that there is an open neighborhood nbd(W) containing W and a smooth function λ such that λ(W) = λW and λ(X) is a simple eigenvalue of X for all X∈nbd(W). We have determined the derivative Dλ(W) and used it to derive three equivalent representations of cond(λ,W). Also, we have constructed a fast perturbation for λW considering a weighted norm on the space of MEPs. We have defined the backward error of an approximate eigenvalue λ of Wby

η(λ,W) := inf{|||∆W|||:λ ∈σ(W+ ∆W)}.

We have derived η(λ,W) and constructed an optimal perturbation ∆W such that λ ∈ σ(W+ ∆W) and |||∆W||| = η(λ,W). Also, for an approximate eigenpair (λ, x :=

x1 ⊗ · · · ⊗ xm) ∈ C^m × (Cⁿ¹ ⊗ · · · ⊗ C^nm), we have defined the backward error by η(λ, x,W) := inf{|||∆W||| : W(λ)x+ ∆W(λ)x = 0}. We have derived η(λ, x,W) and constructed optimal perturbation ∆W for (λ, x) such that W(λ)x+ ∆W(λ)x = 0 and

|||∆W|||=η(λ, x,W). We have extended the results to the case of two parameter PEPs.

Further we have developed a general framework for spectral analysis of linear homoge- neous MEPs. We have introduced condition number cond(λ,W) of a simple eigenvalueλ

of a homogeneous MEP Wand derived cond(λ,W).We have also determined backward errors of approximate eigenvalues and approximate eigenpairs of W and constructed optimal perturbations.

We have defined and analyzed two vector spaces of potential linearizations of two- parameter polynomial of degree k. Considering a two-parameter matrix polynomial of the form

P(λ, µ) :=

Xk i=0

k−i

X

j=0

λⁱµ^jAij,

whereAij ∈C^n×nfor all i= 0,· · · , kandj = 0,· · · , k−i, we have defined and analyzed two vector spaces, namely, the right ansatz space and the left ansatz space of potential linearizations ofP(λ, µ).We have also analyzed conditions under which a pencil in the ansatz spaces is a linearization ofP(λ, µ).

Finally, we have considered structured linear MEPs and analyzed structured back- ward errors and structured backward perturbations. We have denoted the space of structured MEPs by Sand defined structured backward error of an approximate eigen- pair (λ, x:=x₁ ⊗ · · · ⊗x_m) by

η^S(λ, x,W) := inf{|||∆W|||: ∆W∈S and W(λ)x+ ∆W(λ)x= 0}.

We have determined η^S(λ, x,W) and constructed an optimal structured perturbation

∆Wsuch that W(λ)x+ ∆W(λ)x= 0 and|||∆W|||=η^S(λ, x,W).

[1] B. Adhikari, Vector space of linearizations for the quadratic two-parameter matrix polynomial, Linear Multilinear Algebra, 61(2013), pp.603-616.

[2] B. Adhikari, Backward Perturbation and Sensitivity Analysis if Structured Polynomial Eigenvalue Problem, PhD thesis, Department of Mathematics, In- dian Institute of Technology Guwahati, India, 2009.

[3] B. Adhikari and R. Alam, Structured backward errors and pseudospectra of structured matrix pencils, SIAM J. Matrix Anal. Appl., 31(2009), pp.331-359.

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