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Computing a canonical form of a matrix pencil

  • Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits.

    Citation: Miloud Sadkane, Roger Sidje. Computing a canonical form of a matrix pencil[J]. AIMS Mathematics, 2024, 9(5): 10882-10892. doi: 10.3934/math.2024531

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  • Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits.



    Let A,BCn×n, with AλB being a regular matrix pencil, i.e., det(AλB) is not identically zero. If the pencil is regular with respect to the unit circle, i.e., det(AλB)0 for all λ with |λ|=1, then there exists a decomposition (see, e.g., [4, chap. 10]),

    AλB=T[A1λIn100In2λB2]Q,A1Cn1×n1,B2Cn2×n2,n1+n2=n, (1.1)

    in which T,QCn×n are nonsingular, and the eigenvalues of A1 lie inside the open unit disk.

    We are using In to denote the identity matrix of order n. Throughout the paper, we will also be denoting the 2-norm of a given matrix X as X2, while X will stand for its conjugate transpose.

    The decomposition (1.1) is referred to as the canonical form of the pencil AλB. It characterizes that the set of eigenvalues of the pencil AλB is the union of the set of eigenvalues inside the open unit disk (all of which are eigenvalues of A1) and the set of eigenvalues outside the closed unit disk (all of which are reciprocals of the eigenvalues of B2). The decomposition (1.1) is not unique. However, any other decomposition must have the form

    AλB=ˆT[ˆA1λIn100In2λˆB2]ˆQ, (1.2)

    with

    ˆT=T[Y100Y2],ˆQ=[Y1100Y12]Q,ˆA1=Y11A1Y1,ˆB2=Y12B2Y2,

    where Y1 and Y2 are nonsingular matrices. Therefore, in practice, a canonical form can be found only up to such a transformation.

    By using a Möbius transform, it is clear that the unit circle can be replaced by a more general region of the complex plane. In particular, the matrices A1 and B2 can be chosen in Jordan canonical form with a B2 nilpotent (i.e., having only zero eigenvalues), and this yields the Weierstrass canonical form (see, e.g., [14, chap. 6]).

    The spectral projector Pr (resp., Pl) onto the right (resp. left) deflating subspace corresponding to the eigenvalues inside the unit circle can be expressed by the complex contour integral (see, e.g., [4, chap. 10], [6, chap. 4]):

    Pr=12πi|λ|=1(λBA)1Bdλ,Pl=12πi|λ|=1B(λBA)1dλ. (1.3)

    These spectral projectors are connected by the relations PlA=APr, PlB=BPr. They contain all required information on the spectral properties of the pencil AλB. They may be useful, in part or as a whole, for example in model reduction (see, e.g., [2] and references therein), or when analyzing linear differential algebraic systems (see, e.g., [11]).

    Using (1.1), the projectors (1.3) simplify to

    Pr=Q1[In1000]Q,Pl=T[In1000]T1. (1.4)

    Hence, if a decomposition (1.2) is known, the projectors are computable through (1.4). However, from a numerical algorithmic viewpoint, there are other ways to efficiently compute Pr and Pl without resorting to the canonical form. For example by applying a spectral dichotomy method or an inverse-free spectral divide and conquer method [1,10] or methods devised in the context of linear differential algebraic equations [11], although a canonical form –or part of it– may still be required in some circumstances.

    While the knowledge of a canonical form (1.2) immediately implies the knowledge of the projectors through (1.4), it turns out that the converse question has not received much attention. This is the subject of our work: determine the canonical form knowing only the projector Pr or Pl. In other words, solve the reverse problem of going from (1.4) to (1.2). In this case, the factor forms shown in the equalities of (1.4) are not known but are merely identities that presuppose (1.2). Specifically, the newness of the present note is thus: From the projector Pr or Pl, we determine the matrices T, Q, A1 and B2 that characterize the representation (1.1) or the matrices in the equivalent form (1.2).

    As noted earlier, choosing B2 nilpotent results in the Weierstrass canonical form (also called Kronecker canonical form) that generalizes the Jordan canonical form meant for single matrices to regular matrix pairs. It can thus give insights into the structure of a generalized eigenproblem with the matrix pair (A,B) and its underlying applications. As an illustrative example, consider the analysis of a linear system of differential algebraic equations (DAEs) with constant coefficients,

    Bx=Ax+f,

    where B is a singular matrix so that the above cannot simply be multiplied by B1 to revert to ordinary differential equations (ODEs). A canonical form can help transform such DAEs to ease their analysis (e.g., to explore different model parameters or different input vectors f).

    The organization of the paper is as follows. After recalling some past works aimed at computing spectral projectors in Section 2, we give details about our new method in Section 3, followed by examples in Section 4 and finally some concluding remarks in Section 5.

    Given that this study assumes that the spectral projectors are known, we cite here for ease of reference some common approaches used to obtain them in practice. Several spectral dichotomy algorithms credit their roots to the early works of Bulgakov, Godunov and Malyshev, of which [5,10] are examples among many others. In their extensive paper [1], Bai, Demmel, and Ming Gu stated that they built and improved on such early efforts to offer two inverse-free, highly parallel, spectral divide and conquer algorithms: one for computing an invariant subspace of a nonsymmetric matrix and another one for computing the left and right deflating subspaces of a regular matrix pencil. Bai et al. [1] included specifics on how to use a Möbius transform to split the dichotomy along a more general region of the complex plane than the unit circle. Aside from their practical usefulness, the algorithms in [1] were deemed more stable than other parallel algorithms for the nonsymmetric eigenproblem based on the matrix sign function.

    In the course of investigating linear constant coefficient DAEs for properties such as local solvability or asymptotical stability, März [11] indicated the need to decouple those DAEs in canonical form by means of spectral projectors. The work included an iterative algorithm, dubbed matrix and projector chain, producing a sequence of matrices that become stationary and from which the desired projector emerges.

    There are other methods [12] that also retrieve the projector through an iterative process. Finally, we can cite works such as [9,13] which first performed some form of decomposition where eigenvalues come into play (a generalized Schur factorization and block-diagonalization), allowing to ultimately obtain the spectral projections by either solving a system of generalized Sylvester equations or via a special reordering of the Schur factorization.

    We now turn to the newness of our contribution, which is the determination of the canonical form knowing only the projector Pr or Pl. We suppose that Pr is known (a similar reasoning can be applied to Pl and, therefore, is omitted), and denote by

    U1=orth(Pr)andV2=orth(InPr) (3.1)

    the results of a column orthonormalization process that computes the matrices U1 of size n×n1 and V2 of size n×n2 whose columns form orthonormal bases of the range spaces ran(Pr) and ran(InPr). We have the following:

    Theorem 3.1. Assume that the pencil AλB is regular with respect to the unit circle, and consider the canonical form (1.1) with unknown matrices T, Q, A1, and B2. Set Q=[U1V2]1 and T=(A+(BA)Pr)Q1, then the matrices Q and T are nonsingular and we have the canonical form

    AλB=T[A1λIn100In2λB2]Q, (3.2)

    where the eigenvalues of A1 are inside the open unit disk and the reciprocals of the eigenvalues of B2 are outside the closed unit disk; the matrices A1, A1 and B1, B2 are related by

    A1=Y1A1Y11andB2=Y2B2Y12, (3.3)

    where Y1 and Y2 are nonsingular matrices.

    Proof. The orthonormalization process needed in (3.1) can be done (see, for example, [7]) via a QR, SVD (singular value decomposition), or other method. Using, for instance, the SVD, we obtain

    Pr=[U1U2][Σ1000][W1W2]=U1Σ1W1, (3.4a)
    InPr=[V2V1][Γ2000][Z2Z1]=V2Γ2Z2, (3.4b)

    where U1, W1, V1, and Z1 are n×n1 matrices, U2, W2, V2, Z2 are n×n2 matrices, and [U1U2], [W1W2], [V2V1] and [Z2Z1] are unitary matrices. The matrix Σ1 is an n1×n1 diagonal matrix containing the nonzero singular values of Pr and Γ2 is an n2×n2 diagonal matrix containing the nonzero singular values of InPr. Note that n1=trace(Pr)=rank(Pr). The properties P2r=Pr and (InPr)2=InPr imply that

    W1U1Σ1=Σ1W1U1=In1,PrU1=U1, (3.5a)
    Z2V2Γ2=Γ2Z2V2=In2,(InPr)V2=V2. (3.5b)

    From PrU1=U1 and (InPr)V2=V2 we deduce that [In1000]QU1=QU1 and [000In2]QV2=QV2. Therefore QU1 and QV2 can be written as

    QU1=[Y10],QV2=[0Y2],

    where Y1 and Y2 are respectively n1×n1 and n2×n2 nonsingular matrices. As a consequence

    [U1V2]=[PrU1(InPr)V2]=[Q1[In1000]QU1Q1[000In2]QV2]=Q1[Y100Y2].

    This shows that the matrix Q is nonsingular and Q1=Q1[Y100Y2]. A simple calculation yields A+(BA)Pr=TQ, so T is nonsingular and T1=Q(TQ)1. Hence

    T1AQ1=[Y11A1Y100In2],T1BQ1=[In100Y12B2Y2].

    Let the matrices A and B be slightly perturbed to ˜A=A+ΔA and ˜B=B+ΔB, and let the spectral projection Pr be perturbed to ˜Pr=Pr+ΔPr, accordingly. The projection ˜Pr can be expressed as in (1.3) with ˜A and ˜B instead of A and B. Our main objective is to study how far the matrices T and Q (in Theorem 3.1) will change when the matrices A and B are replaced by ˜A and ˜B.

    We assume that the perturbed pencil ˜Aλ˜B is regular and measure the size of perturbations by

    ϵ=max(ΔA2,ΔB2). (3.6)

    Such a measure of perturbations appears when studying the ϵ-pseudosepctrum of the pencil λBA defined by (see, [15])

    Λϵ(A,B)={λ:det(A+ΔAλ(B+ΔB))=0for ΔA,ΔBs.t.max(ΔA2,ΔB2)ϵ}={λ:(λBA)12(1+|λ|)1ϵ}. (3.7)

    A related measure to ϵ is the norm of the resolvent (λBA)1 over the unit circle defined by

    r(A,B)=max|λ|=1(λBA)12. (3.8)

    From (3.7) and (3.8), it is clear that 2r(A,B)1/ϵ whenever λ belongs to both the unit circle and the set Σϵ(A,B). This entails certain restrictions on the existence of the perturbed projection, as the following proposition shows.

    Proposition 1. Let the perturbations ΔA and ΔB be measured as in (3.6) and assume that 2ϵr(A,B)<1, then the perturbed projection ˜Pr is well-defined and the perturbation ΔPr satisfies ΔPr2=O(ϵr2(A,B)).

    Proof. For λ on the unit circle, we have

    λ˜B˜A=λBA+λΔBΔA=(λBA)C(λ),

    where

    C(λ)=(In+(λBA)1(λΔBΔA)).

    Since for |λ|=1, we have

    (λBA)1(λΔBΔA)22ϵr(A,B)<1,

    and we deduce that C(λ) is nonsingular and

    C(λ)12112ϵr(A,B). (3.9)

    We also deduce that λ˜B˜A is nonsingular and, therefore, the projection ˜Pr is well-defined and is given by

    ˜Pr=12πi|λ|=1(λ˜B˜A)1˜Bdλ.

    A straightforward computation now gives

    (λ˜B˜A)1˜B=(λBA)1B+(λBA)1ΔBC(λ)1(λBA)1(λΔBΔA)(λBA)1˜B.

    Therefore

    ˜Pr=Pr+ΔPr,

    where

    ΔPr=12πi|λ|=1((λBA)1ΔBC(λ)1(λBA)1(λΔBΔA)(λBA)1˜B)dλ.

    We have

    ΔPr212π|λ|=1(λBA)12ΔB2|dλ|+|λ|=1(C(λ)12(λBA)122(ΔB2+ΔA2)(B2+ΔB2))|dλ|,

    and using (3.6), (3.8), and (3.9), we deduce that

    ΔPr2ϵr(A,B)1+2r(A,B)B212ϵr(A,B).

    Since 1+2r(A,B)B2=O(r(A,B)), we can write ΔPr2=O(ϵr2(A,B)).

    Proposition 1 shows that the condition 2ϵr(A,B)<1 ensures the existence of ˜Pr, but in order to ensure the nearness of Pr and ~Pr and, hence, the smallness of ΔPr2, we also need ϵr2(A,B) to be small. In particular, the resolvent should not be large on the unit circle. As we will see, the term ϵr2(A,B) essentially represents the error in the perturbed version of Theorem 3.1.

    The following proposition gives an analogue of (3.1) for the perturbed projection.

    Proposition 2. Let U=(In+ΔPr)U1, ˜U1=U(UU)12, V=(InΔPr)V1, and ˜V1=V(VV)12. If ϵ is small enough so that Proposition 1 can be applied, then the columns of ˜U1 (resp., ˜V1) form an orthonormal basis of the range space ran(˜Pr) (resp., ran(In˜Pr)).

    Proof. The assumption on ϵ ensures that Proposition 1 can be applied and, in particular, that In±ΔPr is nonsingular. It follows that

    rank(U)=rank(U1)=rank(˜U1),rank(V)=rank(V1)=rank(˜V1),
    rank((InΔPr)W2)=rank(W2),rank((In+ΔPr)Z2)=rank(Z2).

    From the properties of Pr (see (3.4) and (3.5)), we immediately deduce that

    ˜Pr˜U1=˜U1,(In˜Pr)˜V1=˜V1,
    ˜Pr((InΔPr)W2)=0,(In˜Pr)((In+ΔPr)Z2)=(In+ΔPr)Z2,

    and these properties determine the null space of ˜Pr and In˜Pr.

    Thus, if A and B are slightly perturbed to ˜A=A+ΔA and ˜B=B+ΔB, as in Propositions 1 and 2, then the matrices Q and T in Theorem 3.1 can be replaced by

    ˜Q=[˜U1˜V1]1and˜T=(˜A+(˜B˜A)˜Pr)˜Q1.

    From Proposition 2, it is easy to check that U=U1+O(ϵr2(A,B)), UU=In1+O(ϵr2(A,B)), and so ˜U1=U1+O(ϵr2(A,B)). Similarly, we have ˜V1=V1+O(ϵr2(A,B)). This leads to

    ˜Q=Q+O(ϵr2(A,B)),˜T=T+O(ϵr2(A,B)).

    Two examples are given to illustrate Theorem 3.1. In the first example, the spectral projection is computed numerically by one of the methods referenced earlier, and in the second one, the spectral projection is obtained directly using the spectral properties of the pencil under consideration.

    Example 1. Computations are carried out in MATLAB. Consider

    A=[0000010000100001],B=[2110020010000100].

    The exact eigenvalues of the pencil AλB are 2, 0.5, 0, and .

    The spectal projection Pr onto the right deflating subspace corresponding to the eigenvalues inside the unit circle is computed using the algorithm developed in [1] to which the reader is referred for further details on convergence and stability. Below is a brief description.

    Start with the matrices A and B as initializers for A0 and B0, then proceed iteratively: At each iteration k=0,1,..., perform a QR decomposition of the matrix [BkAk], and replace the matrices Ak and Bk by, respectively, Q12Ak and Q22Bk, where Q12 (resp., Q22) denotes the n×n submatrix of Q ---that results from the QR decomposition--- formed from the first n rows and last n columns (resp., last n rows and last n columns). After a few iterations (see [1, Section 9] for the stopping criterion), a good approximation of the projection Pr is obtained from (Ak+Bk)1Bk. With the matrices of this example, after 10 iterations, we obtain

    Pr=[1.00001.66671010.50010003.33331010000.500].

    Then Theorem 3.1 is used to compute the matrices T, Q, A1 and B2.

    T=[3.68591012.27383.330710161.110210161.33311.07771.830210163.125110166.28681017.77671017.34631015.10211016.66571015.38871015.70431018.2134101]
    Q=[=6.28681011.01213.14341011.062710167.77671016.03851013.88831019.846010189.576010195.913110199.18291015.70431016.650610191.98091016.37761018.2134101]
    A1=[4.42071013.57381017.16611025.7932102],B2=[3.37301012.34261012.34261011.6270101].

    The 2-norm of the absolute error between A, B and the computed canonical form is given by

    AT[A100In2]Q2=8.74111016,BT[In100B2]Q2=1.02711015,

    and the eigenvalues of A1 are 0.5 and 6.93891018 and those of B2 are 0.5 and 0 (having reciprocals 2 and ). Hence, the exact values agree with the numerical predictions of Theorem 3.1.

    Example 2. Consider the matrix pencil AλB with

    A=[KCC0],B=[In000], (4.1)

    where K is an n×n matrix and C is an n×m matrix of full rank with n>m. Such a pencil arises in the stability analysis of the steady-state solutions of equations modeling viscous incompressible flows (see, e.g., [3]). The case when B has the form [M000], where M is a Hermitian positive definite matrix, can be brought back to the matrix B in (4.1) by changing K to L1K(L1) and C to L1C, where L is the Choleski factor of M.

    Consider the QR decomposition of C:

    C=QR=[Q1Q2][R10]=Q1R1, (4.2)

    where R1 is m×m nonsingular and upper triagular, Q is n×n unitary, Q1 is n×m, and Q2 is n×(nm). From [3, Theorem 2.1], the pencil AλB has nm finite eigenvalues and 2m infinite eigenvalues of algebraic multiplicity 2m. The finite eigenvalues are given by the eigenvalues of the matrix Q2KQ2 and the corresponding eigenvectors are spanned by the range of [Q2Q2R11KQ2Q2].

    Let us find the spectral projector Pr corresponding to all finite eigenvalues (instead of the unit circle, we consider here any circle that encloses all finite eigenvalues of the pencil) and the resulting canonical form.

    Using the properties P2r=Pr, BPr=PlB, ran(Pr)=ran([Q2Q2R11KQ2Q2]), and trace(Pr)=rank(Pr)=nm, we deduce that

    Pr=[Q2Q20R11KQ2Q20],In+mPr=[Q1Q10R11KQ2Q2Im],

    and in low rank forms

    Pr=[Q2R11K12][Q20],In+mPr=[0Q1R11R11(K11Im)][Q1KQ1R1Q10],

    where Kij=QiKQj,i,j=1,2. Note that [Q2R11K12] and [0Q1R11R11(K11Im)] have full rank and therefore their columns provide bases for the range spaces of Pr and In+mPr.

    Consider the QR decompositions:

    [Q2R11K12]=U1Y1,[0Q1R11R11(K11Im)]=V2Y2, (4.3)

    where U1 and V2 are matrices whose columns provide orthonormal bases for ran(Pr) and ran(InPr), and Y1 and Y2 are, respectively, n×n and 2m×2m (upper triangular) nonsingular matrices. Using (4.3), the matrices Q and T in Therorem 3.1 are given by

    Q=[U1V2]1=[Y100Y2] [Q20Q1R11K12R11R11(K11Im)]1=[Y100Y2][Q20Q1(KIn)R1Q10]

    and

    T=(A+(BA)Pr)[Q20Q1R11K12R11R11(K11Im)][Y1100Y12]=[Q2Q1Q2K21+Q100R1][Y1100Y12].

    A straightforward computation gives the canonical form stated in Theorem 3.1 with

    A1=Y1K22Y11andB2=Y2[0Im00]Y12,

    thus providing analytic solutions to a class of matrix pencil problems that have the form (4.1).

    It is often convenient in numerical computing to be able to retrieve useful quantities as byproducts from others, as is the case with estimating the residual or condition number when solving a linear system, or obtaining optional factors upon performing a matrix factorization. Here we have described a new method with which to determine the canonical form of a pencil by taking advantage of existing efficient numerical algorithms already developed to determine spectral projections. We performed a perturbation analysis of the proposed method and illustrated it with two examples. The first example was a walk-through that numerically went over the steps of the method, while the second example showed how to further build on the method to theoretically establish an analytic representation of the canonical form of a pencil that arises in the stability analysis of the steady-state solutions of equations modeling viscous incompressible flows. Given that much efforts have been devoted to determining the spectral projections compared to determining the canonical form, our proposed method is a potent bridge from the former to the latter, and it also helps fill a gap in the literature. Application areas where this newfound ability may be useful include model order reduction (see, e.g., [2,8]).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors have declared no conflicts of interest.



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