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Matrix pencils with coefficients that have positive semidefinite Hermitian part

We analyze when an arbitrary matrix pencil is equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with coefficients that have positive semidefinite Hermitian parts. We will make a detailed analysis of their spectral properties and their numerical range. In particular, we relate the Kronecker structure of these pencils to that of an underlying skew-Hermitian pencil and discuss their regularity, index, numerical range, and location of eigenvalues. Further, we study matrix polynomials with positive semidefinite Hermitian coefficients and use linearizations with positive semidefinite Hermitian parts to derive sufficient conditions for a spectrum in the left half plane and derive bounds on the index.

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1 Introduction

In this paper we generalize the class of dissipative Hamiltonian (dH) matrix pencils, which are pencils of the form

 P(λ)=λE−A=λE−(J−R)Q, (1.1)

where , (), , , and . Here stands for the conjugate transpose and (or ) denotes that the Hermitian matrix is positive semidefinite (or positive definite, respectively). Such dH pencils have many favourable properties, e.g. all finite eigenvalues are in the closed left half plane and all purely imaginary eigenvalues are semisimple, except possibly the eigenvalues which may have Jordan blocks of size at most two, see [21]. For a detailed discussion of dissipative and port-Hamiltonian systems and their applications we refer to [1, 4, 5, 11, 23, 28, 29]. The observation that the positivity and symmetry structures of the coefficient matrices of dH pencils lead to these restrictions in the spectrum shows that the imposed structural conditions - although looking rather simple - are in fact very strong. Imposing that the matrices are real leads to several further spectral properties, see [22].

Since spectral properties are invariant under equivalence transformations of the matrix pencil, but the dissipative Hamiltonian structure is not, it is clear that there are many pencils that have the same spectral properties but that do not have the structure of the pencil as in (1.1). As our first result (Theorem 3) we will characterize when a general matrix pencil is equivalent to a dH pencil and we will show that it is necessary and sufficient that the mentioned spectral properties hold.

In several applications matrix pencils arise that carry a structure that is related to, but more general than the one of dH pencils. These are square pencils of the form

 λ(J1+R1)+(J2+R2), with Ji=−J∗i, Ri≥0, i=1,2. (1.2)

In other words, we will assume that the (uniquely defined) Hermitian part of each coefficient is positive semidefinite. We will call these pencils posH pencils, abbreviating ‘positive semidefinite Hermitian part coefficients’. If (or ) then the pencil in (1.2) simply reduces to a dH pencil as in (1.1) with (or its reversal, respectively) and thus, all eigenvalues of the pencil are in the closed left half plane. This is no longer true if both and are nonzero as the following example shows.

Example

Consider the pencil

 λ⎡⎢⎣0−10100001⎤⎥⎦+⎡⎢⎣1000010−10⎤⎥⎦

which has the form (1.2) and is a linearization of the scalar polynomial that has two roots with positive real part.

In view of Example 1 one may initially think that the structure of posH pencils is rather weak compared to that of dH pencils, but we will show in this paper that posH pencils still have many special properties. Furthermore, they are of great importance in applications which makes it necessary to analyze and study them in detail. Let us give two motivating examples.

Example

The space discretization of the Moore-Gibbs-Thompson equation [7, 17] leads to cubic matrix polynomials

 P(λ)=3∑i=0λiAi,

where all coefficients are real symmetric and positive definite. Using structured linearization (see Theorem 6.2 for details) one obtains a pencil of the form (1.2) given by

 λ⎡⎢⎣0−A30A3A2000A0⎤⎥⎦+⎡⎢⎣A3000A1A00−A00⎤⎥⎦. (1.3)

We will analyze under which conditions all eigenvalues of this pencil are in the open left half-plane, see Theorem 6.2 and Corollary 6.2.

Example 1 illustrates that posH pencils arise as linearizations of higher order matrix polynomials with positive (semi-)definite Hermitian coefficients. Further constructions of this type can be found in Remarks 6.1 and 6.1. However, there are other situations that can be modeled with the help of posH pencils.

Example

In the analysis of disk brake squeal, see [13], one has to analyze the spectral properties of quadratic matrix polynomials , with real symmetric positive semidefinite matrices and real skew-symmetric matrices

. Brake squeal is associated to a flutter instability arising at the brake-pad disk interface and it is correlated to eigenvalues with positive real part. Consider the linearization

 λ[M00K−N]+[D+GK+N−K+N0]

that has the form (1.2). If the contribution from the skew-symmetric matrix is zero then this is a dH pencil and if the norm of is sufficiently small, then this pencil still has all eigenvalues in the left half plane. However, if the norm of is larger, then eigenvalues in the right half complex plane occur that may lead to brake squeal.

The three presented examples show that extra assumptions for pencils of the form (1.2) are needed to guarantee that all eigenvalues of such pencils or related matrix polynomials are in the left half plane. We will derive such conditions and also analyze general spectral properties.

The paper is organized as follows. In Section 2 we present some preliminary results and introduce relevant notation. In Section 3 we discuss necessary and sufficient conditions for a pencil to be equivalent to a dH pencil as in (1.1).

In Section 4 we relate the Kronecker structure of a posH pencil of the form to that of the underlying skew-Hermitian pencil with particular emphasis on regularity, the index of the pencil, and positive eigenvalues. The key result here is Theorem 4, which says that the singular part of a posH pencil in (1.2) is contained in the common kernel of and and in the singular part of . This fact leads to several necessary or sufficient conditions for regularity of posH pencils, see Corollaries 4 and 4, and Theorem 5.1.

In Subsection 5.1

we first consider the numerical range for posH pencils, in particular we link the existence of common isotropic vectors with regularity of the pencil, see Theorem

5.1. In Subsection 5.2 we localize the numerical range and the spectrum in a pacman-like shape, the main result is Theorem 5.2. In Subsection 5.3 we provide several sufficient conditions that guarantee that the numerical range or at least the spectrum of a posH pencil is contained in the closed left half plane - a condition that is necessary for stability of the pencil.

In Section 6 we consider the special case of matrix polynomials with positive semidefinite Hermitian coefficients, i.e., the skew-Hermitian parts of the coefficients are all zero. We analyze their index in Theorem 6.1 and localize the spectrum in Theorem 6.2. This is done by showing that these polynomials can be linearized by posH matrix pencils and by applying the results from previous sections.

2 Preliminaries

We denote by the set of matrix polynomials with coefficients in the set of matrices over (). For a pencil the reversal is defined as . Two pencils are called equivalent if there exists invertible matrices , such that . To analyze the spectral properties of matrix pencils we will employ the Kronecker canonical form [10]. Denote by the standard upper triangular Jordan block of size associated with the eigenvalue and let denote the standard right Kronecker block of size , i.e.,

[Kronecker canonical form] Let . Then there exist nonsingular matrices and such that

 S(λE−A)T=diag(Lϵ1,…,Lϵp,L⊤η1,…,L⊤ηq,Jλ1ρ1,…,Jλrρr,Nσ1,…,Nσs), (2.1)

where the parameters are nonnegative integers, , and for as well as for . This form is unique up to permutation of the blocks.

For real matrices a real version of the Kronecker canonical form is obtained under real transformation matrices . In this case the blocks with have to be replaced with corresponding blocks in real Jordan canonical form with diagonal blocks of the form

associated to the corresponding pair of conjugate complex eigenvalues , but the other blocks have the same structure as in the complex case.

An eigenvalue is called semisimple if the largest associated Jordan block has size one. The sizes and of the rectangular blocks are called the left and right minimal indices of , respectively. If is a left minimal index, then there exists a singular chain of vectors satisfying , , and . Similarly, if is a right minimal index, then there exists a singular chain of vectors satisfying , , and . The matrix pencil , is called regular if and for some , otherwise it is called singular. A pencil is singular if and only if it has blocks of at least one of the types or in the Kronecker canonical form.

The values are called the finite eigenvalues of . If , then is said to be an eigenvalue of . (Equivalently, zero is then an eigenvalue of the reversal of the pencil .)

The sum of all sizes of blocks that are associated with a fixed eigenvalue is called the algebraic multiplicity of , while the individual sizes of the Jordan blocks are called the partial multiplicities of . The size of the largest block is called the index of the pencil , where, by convention, if is invertible.

The pencil is called stable if it is regular, if all eigenvalues are in the closed left half plane, and if the ones lying on the imaginary axis (including infinity) are semisimple. Otherwise the pencil is called unstable.

3 Pencils that are equivalent to dH pencils

It is a natural question to ask under which conditions a posH pencil is equivalent to a dH pencil. It turns out that the answer is obtained by a general characterization including matrix pencils without special symmetry and positivity structures.

1. A pencil is equivalent to a pencil of the form as in (1.1) with being regular if and only if the following conditions are satisfied:

1. The spectrum of is contained in the closed left half plane.

2. The finite nonzero eigenvalues on the imaginary axis are semisimple and the partial multiplicities of the eigenvalue zero are at most two.

3. The index of is at most two.

4. The left minimal indices are all zero and the right minimal indices are at most one (if there are any).

2. A pencil is equivalent to a pencil of the form as in (1.1) (i.e., with ) if and only if the following conditions are satisfied:

1. The spectrum of is contained in the closed left half plane.

2. The finite eigenvalues on the imaginary axis (including zero) are semisimple.

3. The index of is at most two.

4. The left and right minimal indices are all zero (if there are any).

The “only if” direction for (i) was proved in [21] and the one for (ii) in [22], see also [12] for the matrix case.

For the “if” direction, we may assume without loss of generality that is in Kronecker canonical form. In particular, we may consider each block separately. First, we prove (ii) and we distinguish the following cases for with .

Case 1): .
Subcase 1a): , , with . Then is equivalent to the pencil , with

By [2, Proposition 2.2] which is a combination of Theorem 2.4 and Proposition 2.5 in [16], it follows that (in fact .
Subcase 1b): with . Here, we have , and .
Subcase 1c): . This pencil is equivalent to and we can take , and .
Subcase 1c’): . Then is equivalent to the pencil

 λE−JQwithE=[0001],J=[01−10],R=0andQ=I2.

Subcase 1d): Since the pencil is square the numbers of left and right minimal indices are equal and hence each pair corresponds in the Kronecker canonical form to a block . Here we can take and .

Case 2): .
Subcase 2a): with . This case works exactly as Subcase 1a) with .
Subcase 2a’): with and . In this case is equivalent to the matrix pencil with

 M = E=Q=I2n, J=12(M−MT),R=−12(M+MT).

Again, by combining Theorem 2.4 and Proposition 2.5 in [16] it follows that .
Subcase 2b): with . Here we have with , , and .
Subcase 2b’): . Here we have and .
The subcases 2c) and 2d) are identical to the subcases 1c) and 1d) as the corresponding matrices are all real.

To prove (i) it remains to consider one additional block of the form in subcase 1b) and one additional combination of minimal indices in subcase 1d). In the first case we have with

 E=I2, R=0, J=[01−10], Q=[0001].

As all matrices are real, this case works for both the real and the complex case.

In the second case, note that again the numbers of left and right minimal indices must be equal. For a pair of left and right minimal indices we are in subcase 1d) of (i). For a pair of left and right minimal indices we have a block

 L(λ)=λ[1000]−[0100]

in the Kronecker canonical form. Here, we can take

 E=[1000],R=0,J=[01−10]andQ=[0001].

Again all matrices are real, so this case works for both the real and the complex case.

Theorem 3 clearly shows that the spectral properties are precisely characterizing the equivalence to matrix pencils of the form (1.1), so we cannot expect similarly nice spectral properties if we generalize to pencils of the form (1.2). However, we still get spectral restrictions for such pencils, some of which are associated with the numerical range which is an important tool in investigating stability of matrices, matrix pencils or matrix polynomials. These will be discussed in the following sections.

4 On the Kronecker structure of posH matrix pencils

In this section we will investigate the Kronecker structure of posH pencils, i.e., matrix pencils of the form (1.2). We have already seen in Example 1 that posH pencils may have eigenvalues in the right half plane including eigenvalues on the positive real axis. In fact, without posing further restrictions on the pencil, any eigenvalue in the complex plane is possible.

Example

Let . If then , i.e., , , , is a complex posH matrix pencil having the eigenvalue . (If then consider the complex posH pencil instead.) In particular, if and , then is an example of a posH pencil with an eigenvalue on the positive real axis.

Example

For an example with real matrix coefficients consider the posH matrix pencil with

 R1=0,J1=[01−10],J2=[0α−α0]andR2=[β00β],

where and . Then has a pair of conjugate complex eigenvalues . In particular, if and , then has a double eigenvalue on the positive real axis.

Although the spectrum may contain any value of the complex plane, not any Kronecker structure is possible for posH pencils. In the following we will discuss restrictions on the index and the structure of the singular part of such pencils. We start with two technical results on values and vectors satisfying . Note that the pencil is not excluded to be singular, so is not necessarily an eigenvalue of .

Let be a pencil as in (1.2) and let and .

1. If , and , then

 R1x=R2x=0and(λ0J1+J2)x=0.
2. If , and then

 R1x=R2x=0 andx∗J1x=0=x∗J2x.

Let with . First observe that in both cases (i) and (ii). Taking the real and imaginary parts independently yields the equations

 αx∗R1x+iβx∗J1x = −x∗R2x, (4.1) αx∗J1x+iβx∗R1x = −x∗J2x (4.2)

(i) Assume and , then we obtain from (4.1) that which, by the semidefiniteness of and , is only possible if . But then we have .

(ii) Let be arbitrary. Then due to one has and , which implies . Hence, thanks to (4.1) we have and furthermore by (4.2).

By Theorem 3 the left and right minimal indices of a singular dH pencil with as in (1.1) can only be zero. This is no longer true for posH pencils of the form (1.2), but the following result shows that the singular part of posH pencils is still restricted. Let be a pencil of the form (1.2). If is a singular chain associated with a left or right minimal index , then and is also a singular chain of associated with a left respectively right minimal index . Let be a singular chain associated with a left or right minimal index of . Without loss of generality, let this be a right minimal index, otherwise, consider the pencil with coefficients that are the conjugate transposes of that of . Then we have

 (J1+R1)x1=0,(J1+R1)xi+1=(J2+R2)xi,i=1,…,k,(J2+R2)xk+1=0, (4.3)

or equivalently, using that and and multiplying by ,

 x∗1(J1−R1)=0,x∗i+1(J1−R1)=x∗i(J2−R2),i=1,…,k,x∗k+1(J2−R2)=0. (4.4)

We first prove by induction that for all . From we get and thus . If , then we have and also follows similarly from . Otherwise we have

 x∗1(J2+R2)x1=x∗1(J1+R1)x2=0

which implies that .

Suppose that for some we have shown for all . If then we are done, because similar to the previous argument we then get from and from .

Hence, we may assume that and thus . Using (4.3) and (4.4) we obtain

 x∗ℓ+1(J1+R1)xℓ+1=x∗ℓ+1(J2+R2)xℓ=x∗ℓ+1(J2−R2)xℓ=x∗ℓ+2(J1−R1)xℓ=x∗ℓ+2(J1+R1)xℓ,

where we have used that and . We repeat this procedure times, obtaining , and we may proceed until . If , i.e., if , then we have

 x∗ℓ+1(J1+R1)xℓ+1=x∗ℓ+1+m(J1+R1)x1=0,

while if , then we get

 x∗ℓ+1(J1+R1)xℓ+1=x∗k+1(J1+R1)xℓ+1−m=x∗k+1(J2+R2)xℓ−m=0.

Thus, in both cases, we finally obtain which implies that .

On the other hand, using (4.3), (4.4) and that we just proved that and hold, we obtain that

 x∗ℓ+1(J2+R2)xℓ+1 = x∗ℓ+1(J1+R1)xℓ+2=x∗ℓ+1(J1−R1)xℓ+2 = x∗ℓ(J2−R2)xℓ+2=x∗ℓ(J2+R2)xℓ+2.

As before, we repeat this step times, obtaining , and we proceed until . If , then

 x∗ℓ+1(J2+R2)xℓ+1=x∗ℓ+1−m(J2+R2)xk+1=0,

otherwise we have , which gives

 x∗ℓ+1(J2+R2)xℓ+1=x∗1(J2+R2)xℓ+1+m=x∗1(J1+R1)xℓ+2+m=0.

In both cases, we obtain which implies that .

Thus, using an induction argument, we obtain for all . Inserting that into (4.3), we get

 J1x1=0,J1xi+1=J2xi,i=1,…,k,J2xk+1=0

which shows that is a singular chain of the pencil associated with the right minimal index .

Since a pencil of skew-Hermitian matrices has equal left and right minimal indices, see [27], we immediately obtain by Theorem 4 that the same is true for posH pencils.

Let be a pencil of the form (1.2). Then the ordered lists of left and right minimal indices of coincide.

In [22] it was shown that a dH pencil of the form (1.1) (with ) is singular if and only if the three matrices , , and have a common kernel. A corresponding result for posH pencils is only true under additional assumptions. Let be a pencil of the form (1.2). If and if all minimal indices of are zero, then is singular if and only if the four matrices have a common kernel. Moreover, in this case all left and right minimal indices of are zero. This is a direct consequence of Theorem 4.

In fact, Corollary 4 is a direct generalization of the corresponding result on dH pencils. Indeed, if , then the pencil can only have left and right minimal indices equal to zero and hence, the same is true for any pencil of the form with Hermitian positive semidefinite and , i.e., a pencil of the form as in (1.1) with , see part (v) of [22, Theorem 2].

The latter result on dH pencils can also be generalized to posH pencils in a different way by considering other combinations of three of the four coefficients. Furthermore, by considering pencils built of two of the four coefficients of posH pencils, one can characterize situations when pencils of the form (1.2) may or may not have positive real eigenvalues. Let be a pencil of the form (1.2).

1. If is singular then the matrices in each triple , , have a common kernel.

2. If the pencil is regular, then is regular and has no eigenvalues on the real positive axis.

3. If the pencil is regular, then is regular and every real positive eigenvalue of is also an eigenvalue of the pencil .

4. If the pencil is regular, then is regular and if

is an eigenvector associated with a real positive eigenvalue

of then .

5. If the pencil is regular, then is regular and if is an eigenvector associated with a real positive eigenvalue of then .

(i) follows directly from Theorem 4 and (ii) follows from Lemma 4 and (i). To prove (iii) assume that is singular. Then for any there exists a nonzero with and by Lemma 4 i) then as well. Hence is singular. The second claim then follows directly. To see (iv) let be a singular pencil, then by (i) the matrices and have a common kernel and the pencil is singular. The second statement of (iv) follows now directly from Lemma 4. The proof for (v) is analogous to that for the case (iv).

Remark

The canonical forms for real or complex skew-Hermitian pencils are well-known and given in [27]. These canonical forms show that in the real case all eigenvalues of a skew-symmetric matrix pencil necessarily have even algebraic multiplicity. This explains why the real positive eigenvalue of the real posH pencil in Example 4 is a double eigenvalue.

So far, we have discussed the regularity of posH pencils as well as conditions when the spectrum does not intersect the positive real line. Next, we will study the index of such pencils. Although the index may be as large as the size of the pencil, we have the following relation to the underlying skew-Hermitian matrix pencil. Let be a pencil of the form (1.2), let be a Jordan chain of length of associated with the eigenvalue and let .

Then we have as well as and if is even, then also . The Jordan chain of associated with the eigenvalue satisfies

 (J1+R1)x1=0,(J1+R1)xi+1=(J2+R2)xi,i=1,…,k−1,(J2+R2)xk≠0 (4.5)

or equivalently, using that and and multiplying by ,

 x∗1(J1−R1)=0,x∗i+1(J1−R1)=x∗i(J2−R2),i=1,…,k−1,x∗k(J2−R2)≠0. (4.6)

For the remainder of the proof we use a strategy similar to the one in the proof of Theorem 4. From we get and thus . Furthermore, we have

 x∗1(J2+R2)x1=x∗1(J1+R1)x2=0

which implies that .

Suppose that for some with we have shown for all . Then we have . Using (4.5) and (4.6) we obtain

 x∗ℓ+1(J1+R1)xℓ+1=x∗ℓ+1(J2+R2)xℓ=x∗ℓ+1(J2−R2)xℓ=x∗ℓ+2(J1−R1)xℓ=x∗ℓ+2(J1+R1)xℓ,

where we have used that and . We repeat this procedure times, obtaining

 x∗ℓ+1(J1+R1)xℓ+1=x∗2ℓ+1(J1+R1)x1=0

which implies . If and

is odd then we are done. Otherwise (i.e.

or is even) we have . Then using (4.5), (4.6), and that we just proved that holds, we obtain that

 x∗ℓ+1(J2+R2)xℓ+1 = x∗ℓ+1(J1+R1)xℓ+2=x∗ℓ+1(J1−R1)xℓ+2 = x∗ℓ(J2−R2)xℓ+2=x∗ℓ(J2+R2)xℓ+2.

As before, we repeat this step times, obtaining

 x∗ℓ+1(J2+R2)xℓ+1=x∗1(J2+R2)x2ℓ+1=x∗1(J1+R1)x2ℓ+2=0.

This implies implies . Finally, the claim follows using an induction argument.

Let be a pencil of the form (1.2) and assume that the pencil has at most index and right minimal indices that are at most . Then the index of is at most .

Comparing the proofs of Theorem 4 and Theorem 4, we see that the main difference is that in the proof of Theorem 4 we can no longer use the identity , but only . This requires us to “push through” the chains to the first vector instead of possibly to the last vector . This leads to the fact that not necessarily all vectors of the chain are in the joint kernel of the matrices and . The following examples show that the bound given in Theorem 4 is sharp.

Example

Consider the pencil with ,

 J1=⎡⎢⎣0000010−10⎤⎥⎦,J2=⎡⎢⎣001000−100⎤⎥⎦,R2=⎡⎢⎣000010000⎤⎥⎦.

Then