Hermitian EGFPRs preserving the sign characteristic

Hermitian matrix polynomials arise in many applications, and the sign characteristics of their real eigenvalues play an important role, see [1, 7, 11, 12, 16, 33, 34, 43, 44, 47, 48] and the references therein. For the rest of this section we assume that P(λ) = Pm

j=0λ^jA_j is Hermitian with A_m being nonsingular. We characterize a subclass of Hermitian EGFPRs which preserve the sign characteristic of P(λ).

WhenP(λ) is Hermitian, the Jordan formJ ofP(λ) is a direct sum of Jordan blocks associated with real eigenvalues and blocks of the type diag(J_r, J_r), whereJ_r is an r×r Jordan block associated with a nonreal eigenvalue λ and J_r is a Jordan block of the same size associated with λ, see [1, 16, 33, 34] for details. For convenience we use the following notation. By Jr(λ) we denote the Jordan block associated with the eigenvalue λ of size r×r if λ is real, and the direct sum of two Jordan blocks of sizer/2×r/2, if λ is not real, in which case the first block corresponds toλ and the second toλ [34, 16].

To define the sign characteristic of P(λ) we need the following notations.

Following [16] we define

C_P :=























0 I_n 0 · · · 0

0 0 I_n 0

... . .. ...

... I_n

−A⁻¹_m A₀ −A⁻¹_m A₁ · · · −A⁻¹_m Am−1























and B_P :=

















A₁ A₂ · · · A_m A2 ... · ··

... A_m

Am 0















 .

Note that λI_mn−C_P is a strong linearization of P(λ). For an integerk ≥1,we define

Rk:=











1

· ·· 1











∈C^k×k.

Theorem 4.2.1 ([16], Theorem 2.2). Let P(λ) be a Hermitian matrix polynomial with nonsingular leading coefficient. Let λ₁, . . . , λ_α be the real eigenvalues of P(λ) and λα+1, . . . , λβ be the nonreal eigenvalues of P(λ) from the upper half-plane. Then there

exists a nonsingular matrix H such that

J :=H⁻¹C_PH =J_{`1</sub>(λ<sub>1</sub>)⊕ · · · ⊕J<sub>`α}(λ_α)⊕J_{`α+1</sub>(λ<sub>α+1</sub>)⊕ · · · ⊕J<sub>`β}(λ_β) and

P_,J :=H^∗B_PH=₁R_{`1</sub> ⊕ · · · ⊕<sub>α</sub>R<sub>`α} ⊕R_{`α+1</sub>⊕ · · · ⊕R<sub>`β},

where ={₁, . . . , _α} is an ordered set of signs ±1. The set is unique up to permu- tation of signs corresponding to Jordan blocks.

Definition 4.2.2. [16] Let P(λ) be as in Theorem 4.2.1. The set {1, . . . , α} in Theo- rem 4.2.1 is called the sign characteristic of P(λ).

In the special case when λ₀ is a simple eigenvalue of P(λ), the sign in the sign characteristic of P(λ) corresponding to λ₀ is given by sign(x^∗P⁰(λ₀)x), where x is an eigenvector of P(λ) associated with λ₀ and P⁰(λ) is the first derivative of P(λ), see [1, 16, 34].

Recall that a matrix assignment X is said to be a Hermitian matrix assignment if all the matrices belonging to X are Hermitian. Recall that, if P(λ) is Hermitian and the matrix assignments X and Y are Hermitian, then the EGFPR

LP(h,tw,tv,X,Y) =

M_tv(Y)M_tw(X) λM_v^P −M_w^P

M_c^P_wM_rev(tw)(rev(X))M_c^P_vM_rev(tv)(rev(Y)) (4.3) given in Theorem 4.1.17 is Hermitian. We refer the pencil (4.3) as a Hermitian EGFPR of P(λ). In particular, whent_w =∅=t_v, then the Hermitian EGFPR L_P(h,t_w,t_v,X,Y) is denoted byLP(h,∅,∅), that is, LP(h,∅,∅) := λM_v^P −M_w^P

M_c^P_wM_c^P_v.

Caution: As already mentioned in Definition 4.1.15 we always consider nonsingular matrix assignmentsX and Y in the Hermitian EGFPR L_P(h,t_w,t_v,X,Y).

Definition 4.2.3. [44, 16] Two pencils λX+Y and λXe+Ye are said to be ^∗congruent if there exists a nonsingular matrix Q such that λX+Y =Q(λXe+Ye)Q^∗.

Lemma 4.2.4. [44, 16] Let L(λ) = λL₁ −L₀ and Lb = λbL₁ −Lb₀ be two complex Hermitian pencils such that L₁ and Lb₁ are nonsingular. Then L(λ) and L(λ)b have the same elementary divisors and the same sign characteristic if and only if L(λ) and L(λ)b are *congruent.

The following result will be useful for characterizing a subclass of the Hermitian EGFPRs of P(λ) that preserves the sign characteristic of P(λ).

Lemma 4.2.5. Let LP(h,tw,tv,X,Y) and LP(h,∅,∅) be Hermitian EGFPRs of P(λ) as in (4.3). Then LP(h,tw,tv,X,Y) is *congruent to LP(h,∅,∅). More precisely, L_P(h,t_w,t_v,X,Y) =QL_P(h,∅,∅)Q^∗, where Q:=M_tv(Y)M_tw(X) is nonsingular.

Proof. Recall that t_w and c_v commute. Hence we have L_P(h,t_w,t_v,X,Y) = M_tv(Y)M_tw(X) λM_v^P −M_w^P

M_c^P_wM_rev(tw)(rev(X))M_c^P_vM_rev(tv)(rev(Y))

=M_tv(Y)M_tw(X) λM_v^P −M_w^P M_c^P

wM_c^P

vM_rev(tw)(rev(X))M_rev(tv)(rev(Y))

=M_tv(Y)M_tw(X)L_P(h,∅,∅)M_rev(tw)(rev(X))M_rev(tv)(rev(Y)).

Let Z be an n×n arbitrary Hermitian matrix. Then (M_i(Z))^∗ =M_i(Z^∗) =M_i(Z) for i=±0,±1, . . . ,±m. As X and Y are Hermitian matrix assignments, we have

(

M_tv(Y)M_tw(X)

)

^∗ =

(

M_tw(X)

)

^∗

(

M_tv(Y)

)

^∗ =M_rev(tw)(rev(X))M_rev(tv)(rev(Y)).

Hence the desired conclusion follows asX andY are nonsingular matrix assignments.

The following result characterizes a subclass of the Hermitian GFPRs which preserves the sign characteristic of P(λ).

Theorem 4.2.6 ([16], Theorem 7.1). Let h be an even integer such that0≤h≤m−1, and let T(λ) =L_P(h,t_w,t_z, X, Y) be a Hermitian GFPR for P(λ), where X and Y are nonsingular Hermitian matrix assignments for t_w and t_z, respectively. Then T(λ) is a Hermitian strong linearization of P(λ) that preserves the sign characteristic of P(λ).

Thus, in particular, a Hermitian GFPR L_P(h,∅,∅) is a Hermitian strong linearization of P(λ) that preserves the sign characteristic of P(λ).

The following result will be useful for characterizing a subclass of the Hermitian EGFPRs of P(λ) that preserves the sign characteristic of P(λ).

Proposition 4.2.7. Let 0≤ a < b < d≤ m−1, where a is even, and {a+ 1 :b} and {b+1 :d}contain odd number of elements. Let −(m+a+1)+xbe the simple admissible tuple of {a+ 1 :m}, y be the simple admissible tuple of {b+ 1 :d}, −(a+b+ 1) +u be the simple admissible tuple of {a+ 1 :b} and −(m+d+ 1) +sbe the simple admissible tuple of {d+ 1 :m}. Then we have

M_y^PM_x^P =M_u^PM_s^P and M_c^P_xM_rev(y)^P =M_c^P_yM_c^P_uM_c^P_s, (4.4) where c_x, c_y c_u and c_s are the symmetric compliments of x, y, u and s, respectively.

Proof. Given that a is even. Since {a+ 1 : b} and {b+ 1 : d} contain odd number of elements, we have that b is odd andd is even. As y, u and s are the simple admissible tuples, we have

y =

d−1 :d, d−3 :d−2, . . . , b+ 2 : b+ 3, b+ 1 ,

u =

−(a+ 2) : −(a+ 1),−(a+ 4) :−(a+ 3), . . . ,−(b−1) : −(b−2),−b , and s =







−(d+ 2) :−(d+ 1), . . . ,−(m−1) : −(m−2),−m

if m is odd −(d+ 2) :−(d+ 1), . . . ,−(m−2) : −(m−3),−m :−(m−1)

if m is even.

Further, we have x= (α, β, γ), where α :=

−(a+ 2) :−(a+ 1),−(a+ 4) :−(a+ 3), . . . ,−(b−1) : −(b−2)

,

β :=

−(b+ 1) :−b,−(b+ 3) :−(b+ 2), . . . ,−(d−2) :−(d−3),−d:−(d−1) , and γ :=







−(d+ 2) :−(d+ 1), . . . ,−(m−1) : −(m−2),−m

if m is odd −(d+ 2) :−(d+ 1), . . . ,−(m−2) : −(m−3),−m :−(m−1)

if m is even.

Observe thatu= (α,−b) ands=γ. Hencex= (α, β,s).A straight forward calculation shows that M_y^PM_β^P =M_−b^P . Thus we have

M_y^PM_x^P = M_y^PM_α^PM_β^PM_s^P =M_α^PM_y^PM_β^PM_s^P asα and ycommute

= M_α^PM_−b^P M_s^P =M_(α,−b)^P M_s^P =M_u^PM_s^P which yields the first part of (4.4).

Next, we show that M_c^P_xM_rev(y)^P =M_c^P_yM_c^P_uM_c^P_s. Asc_x is the symmetric complement of x, we have

c_x=

−(a+ 2),−(a+ 4), . . . ,−(b−1),−(b+ 1),−(b+ 3), . . . ,

−(d−2),−d,−(d+ 2),−(d+ 4), . . . ,−(m^∗−2),−m^∗ ,

where m^∗ :=m if m is even and m^∗ :=m−1 if m is odd. Similarly, as c_y, c_u and c_s are the symmetric complements of y,u and s, respectively, we have

cy =

(

d−1, d−3, . . . , b+ 4, b+ 2

)

,

c_u =

(

^{−(a+ 2),−(a}+ 4), . . . ,−(b−3),−(b−1)

)

^{, and}

c_s =

(

−(d+ 2),−(d+ 4), . . . ,−(m^∗−2),−m^∗

)

,

where m^∗ :=mif m is even and m^∗ :=m−1 if m is odd. Observe thatcx = cu, δ,cs

, whereδ :=

(

−(b+ 1),−(b+ 3), . . . ,−(d−2),−d

)

.A straight forward calculation shows that M_δ^PM_rev(y)^P =M_c^P

y. This implies that M_c^P_xM_rev(y)^P = M_c^P_uM_δ^PM_c^P_sM_rev(y)^P

= M_c^P_uM_δ^PM_rev(y)^P M_c^P_s as yand cs commute

= M_c^P_uM_c^P_yM_c^P_s =M_c^P_yM_c^P_uM_c^P_s as c_y and c_u commute.

This completes the proof.

The following result is a corollary to Theorem 4.1.17.

Corollary 4.2.8. Let L(λ) :=L_P(h,t_w,t_v,X,Y) be a Hermitian EGFPR of P(λ)such that h₁ is even, where h={h₁, h₂, . . . , h_k}. Then L(λ) is a Hermitian strong lineariza- tion of P(λ).

Proof. Since h₁ is even, it follows from Remark 4.1.13 that 0 ∈/ c_w. Given that P(λ) is Hermitian with nonsingular Am and the matrix assignments X and Y are nonsingular.

Consequently, it follows by Theorem 4.1.17 thatL(λ) is a Hermitian strong linearization of P(λ).

The next result characterizes a subclass of the Hermitian EGFPRs which preserves the sign characteristic of P(λ).

Theorem 4.2.9. Let h = {h1, h2, . . . , hk}, where k ≥ 1 is odd, 0 ≤ h1 < h2 < · · · <

h_k−1 < h_k ≤m−1with evenh₁. LetL(λ) :=L_P(h,t_w,t_v,X,Y)be a Hermitian EGFPR of P(λ), where X and Y are nonsingular Hermitian matrix assignments for t_w and t_v, respectively. Then L(λ) is a Hermitian strong linearization of P(λ) that preserves the sign characteristic of P(λ).

Proof. Let T(λ) := (λM_z^P −M_w^P₁)M_c^P_w

1M_c^P_z, where w₁ is the simple admissible tuple of {0 : h₁}, −(m +h₁ + 1) + z is the simple admissible tuple of {h₁ + 1 : m}, and c_w1 and c_z are the symmetric compliments of w₁ and z, respectively. Note that T(λ) is a Hermitian GFPR of P(λ). Hence by Theorem 4.2.6, T(λ) is a Hermitian strong linearization of P(λ) that preserves the sign characteristic of P(λ). We will prove that L(λ) is *congruent to T(λ), which together with Corollary 4.2.8 and Lemma 4.2.4 will give the desired result.

By Lemma 4.2.5,L(λ) is *congruent toL_P(h,∅,∅) = λM_v^P −M_w^P

M_c^P_wM_c^P_v. Hence we only need to show that L_P(h,∅,∅) is *congruent to T(λ) = (λM_z^P −M_w^P₁)M_c^P_w

1M_c^P_z. Note that L_P(h,∅,∅) = λM_v^P − M_w^P

M_c^P_wM_c^P_v, where w = (w₁,w₃, . . . ,w_k), v =

(v2,v4, . . . ,vk+1), cw = (cw1,cw3, . . . ,cw_k) and cv = (cv2,cv4, . . . ,cv_k+1). We show that QT(λ)Q^∗ =LP(h,∅,∅), i.e.,

Q(λM_z^P −M_w^P₁)M_c^P_w

1M_c^P_zQ^∗ = (λM_v^P −M_w^P)M_c^P_wM_c^P_v, (4.5) whereQ:=M_w^P

3,w5,...,wk. It is enough to prove thatM_c^P

w1M_c^P

zQ^∗ =M_c^P

wM_c^P

v andQM_z^P = M_v^P which would imply (4.5). In other words, we show that

M_w^P_3,w5,...,wkM_z^P =M_v^P_{2,v4,...,vk+1} and (4.6) M_c^P_w

1M_c^P_zM_rev(w^P _3,w5,...,w

k) =M_(c^P_w

1,cw3,...,c_wk)M_(c^P_v

2,cv4,...,c_vk+1). (4.7) If k = 1 then L(λ) = T(λ) and the desired result follows trivially. So we consider thatk > 1. Note thatk is odd. Sinceh₁ is even and by Definition 4.1.11,{h_j−1+ 1 :h_j} contains odd number of elements for each j = 2 : k, we have h₁, h₃, . . . , h_k are even and h₂, h₄, . . . , hk−1 are odd. As −(m+h₁ + 1) +z is the simple admissible tuple of {h₁+ 1 :m}, we have z=



















−(h1+ 2) :−(h1+ 1),−(h1+ 4) :−(h1+ 3), . . . ,

−(m−1) :−(m−2),−m







if m is odd −(h₁+ 2) :−(h₁+ 1),−(h₁+ 4) :−(h₁+ 3), . . . ,

−(m−2) : (m−3),−m:−(m−1)







if m is even We prove (4.6) and (4.7) by induction on number k of elements in the partition h. If k = 3 (i.e., h = {h₁, h₂, h₃}). Then L_P(h,∅,∅) = λM_v^P −M_w^P

M_c^P_wM_c^P_v, where w = (w₁,w₃), v = (v₂,v₄), c_w = (c_w1,c_w3) and c_v = (c_v2,c_v4). The desired result follows from Proposition 4.2.7 by taking x=z, y=w₃, u=v₂,s=v₄,a =h₁,b =h₂ and d=h₃.

Assume that (4.6) and (4.7) hold for any partition h⁰ of {0 :m}that contains k−2 elements, i.e., h⁰ ={h⁰₁, h⁰₂, . . . , h⁰_k−2}, where 0≤h⁰₁ < h⁰₂ <· · ·< h⁰_k−2 ≤m−1 withh⁰₁ even. Now we prove the result forh={h₁, h₂, . . . , h_k}. Define h⁰_j :=h_j forj = 1 : k−2 and h⁰ :={h⁰₁, h⁰₂, . . . , h⁰_k−2}. Then

L_P(h⁰,∅,∅) = (λM_v^P0−M_w^P0)M_c^P

w0M_c^P

v0,

wherew⁰ = (w⁰₁,w⁰₃, . . . ,w⁰_k−2),v⁰ = (v⁰₂,v⁰₄, . . . ,v⁰_k−1) andc_w⁰ andc_v⁰are the symmetric complements of w⁰ and v⁰, respectively. Observe that w⁰_j =w_j for j = 1,3, . . . , k −2, and v⁰_j = v_j for j = 2,4, . . . , k−3. This imply that c_w⁰

j = c_wj for j = 1,3, . . . , k −2, and c_v⁰

j =c_vj for j = 2,4, . . . , k−3. Thus L_P(h⁰,∅,∅) = λM_(v^P

2,v4,...,vk−3,v⁰_k−1)−M_(w^P

1,w3,...,wk−2)

M_(c^P_w

1,cw3,...,c_wk−2)M_(c^P_v

2,cv4,...,c_vk−3,c_v0 k−1),

where −(m+hk−2+ 1) +v⁰_k−1 is the simple admissible tuple of {hk−2+ 1 :m}. Now by induction hypothesis, we have

M_w^P_{3,w5,...,wk−2}M_z^P =M_v^P0

2,v⁰₄,...,v⁰_k−3M_v^P0

k−1 =M_v^P_{2,v4,...,vk−3}M_v^P0

k−1 and M_c^P_w

1M_c^P_zM_rev(w^P

3,w5,...,wk−2) = M_(c^P

w0 1,c_w0

3,...,c_w0

k−2) M_(c^P

v0 2,c_v0

4,...,c_v0 k−3,c_v0

k−1)

= M_(c^P

w1,cw3,...,c_wk−2) M_(c^P

v2,cv4,...,c_vk−3,c_v0 k−1). This implies that

M_w^P_{3,w5,...,wk−2,w}

kM_z^P =M_v^P_{2,v4,...,vk−3}M_w^P

kM_v^P0

k−1 and M_c^P_w

1M_c^P_zM_rev(w^P _{3,w5,...,wk−2,wk)} =M_(c^P_w

1,cw3,...,c_wk−2)M_(c^P_v

2,cv4,...,c_vk−3)M_c^P

v0 k−1

M_rev(w^P _k). In view of (4.6) and (4.7), we only need to show that

M_w^P

kM_v^P0

k−1 =M_v^P_k−1M_v^P

k+1 and M_c^P

v0 k−1

M_rev(w^P

k) =M_c^P

wkM_c^P

vk−1M_c^P

vk+1. (4.8) Note thatw_kis the simple admissible tuple of{h_k−1+ 1 :h_k},−(h_k−2+h_k−1+ 1) +v_k−1 is the simple admissible tuple of{hk−2+ 1 :hk−1}and−(m+h_k+ 1) +v_k+1 is the simple admissible tuple of {h_k+ 1 :m}. Thus the desired result follows from Proposition 4.2.7 by taking x=v⁰_k−1,y=w_k, u=vk−1, s=v_k+1, a=hk−2, b =hk−1 and d=h_k. Example 4.2.10. Let P(λ) := P7

i=0λⁱA_i be Hermitian, where A₇ is nonsingular. Let h := {h1, h2, h3}, where h1 = 0, h2 = 3 and h3 = 6. Let tw = (4), tv = (−3), and X and Y be any Hermitian matrices. Then by Theorem 4.2.9, L_P(h,t_w,t_v, X, Y) = M₋₃(Y)M₄(X)

(

λM(−7,−2,−1,−3)^P −M_(0,5,6,4)^P

)

M₅^PM₄(X)M₋₂^P M₋₃(Y)

=



































λA7+A6 A5 −X 0 −In 0 0

A₅ −λA₅+A₄ λX−I_n 0 λI_n 0 0

−X λX−In 0 0 0 λIn 0

0 0 0 0 0 −I_n λI_n

−In λIn 0 0 0 −Y λY

0 0 λI_n −I_n −Y λA₃−A₂ λA₂

0 0 0 λIn λY λA2 λA1 +A0



































is a Hermitian strong linearization of P(λ) and preserves the sign characteristic of P(λ).

We end this section with a discussion on the construction of low band-width banded Hermitian EGFPRs preserving the sign characteristic of P(λ). Recall that there does not exist any block penta-diagonal Hermitian GFPR of P(λ) when P(λ) has degree m ≥ 8. This implies that there does not exist any block penta-diagonal Hermitian GFPR which preserves the sign characteristic of P(λ) when m ≥ 8. Further, T(λ) = (λM(−5:−4,−7:−6)^P −M_(2:3,0:1)^P )M_(2,0)^P M_(−5,−7)^P is the only block penta-diagonal Hermitian GFPR of P(λ) when m = 7. It is shown in [16, p. 266] that a Hermitian GFPR L_P(h₁,t_w,t_z,X,Y) with odd h₁ is a strong linearization of P(λ) preserving the sign characteristic ofP(λ) if and only if the Hermitian GFPRL_P(1,∅,∅) and the Hermitian GFPR L_P(0,∅,∅) are *congruent, which is not true for all Hermitian P(λ), see [16, Example 7.6]. Thus, in general, T(λ) does not preserves the sign characteristic of P(λ). Consequently, there does not exist any block penta-diagonal Hermitian GFPR which preserves the sign characteristic of P(λ) when m ≥ 7. On the other hand, the Hermitian EGFPR given in Example 4.2.10 is a strong linearization and preserves the sign characteristic of a Hermitian P(λ). Moreover, we have the following result.

Corollary 4.2.11. LetL(λ) :=L_P(h,t_w,t_v,X,Y)be the pencil given in Theorem 4.1.21 such that h₁ is even (i.e., h₁ ∈ {0,2}) and X and Y are nonsingular Hermitian matrix assignments for t_w and t_v, respectively. Then L(λ) is a Hermitian block penta-diagonal strong linearization of P(λ) which preserves the sign characteristic of P(λ).

Proof. The desired result follows from Theorem 4.1.21 and Theorem 4.2.9.

Recall that there does not exist any block tridiagonal Hermitian GFPR of P(λ) when P(λ) has degree m ≥ 4. Further, (λM_(−3,−2)^P −M_(0,1)^P )M₀^PM₋₃^P is the only block tridiagonal Hermitian GFPR of P(λ) when m = 3. Now, by similar arguments as in the case of block penta-diagonal Hermitian GFPR it follows that there does not exist any block tridiagonal Hermitian GFPR which preserves the sign characteristic of P(λ) when m≥3. For EGFPRs, we have the following result.

Corollary 4.2.12. Consider the EGFPR L(λ) := L_P(h,t_w,t_v,X,Y) as given in The- orem 4.1.23 such that h₁ = 0. Then L(λ) is a Hermitian block tridiagonal strong linearization of P(λ) which preserves the sign characteristic of P(λ).

Proof. The desired result follows from Theorem 4.1.23 and Theorem 4.2.9.

For example, (λM_{(−4,−3,−1)}^P −M_(0,2)^P )M₋₄^P is a Hermitian EGFPR that preserves the sign characteristic P(λ) whenP(λ) is Hermitian with degree 4.