ISSN 1064�5624, Doklady Mathematics, 2013, Vol. 88, No. 2, pp. 545–547. © Pleiades Publishing, Ltd., 2013.Original Russian Text © N.F. Valeev, 2013, published in Doklady Akademii Nauk, 2013, Vol. 452, No. 4, pp. 359–362.

545

1. We define and analyze a new spectral characteristicof pencils of linear operators acting in the n�dimensionalEuclidean space �n. Before formulating the problem,we give some preliminaries (see, e.g., [1, 5, 9, 10])related to pencils of the form

(1)

The normal rank of a pencil A – μB is defined as

(2)

Since rank(A – μB) = rank , the rank of A –

μB at the point μ = ∞ is equal to rank(B).A pencil A – μB: �n → �n is said to be regular if

nrank(A – μB) = n. If nrank(A – μB) < n, then the pencilis called singular.

A number μ* ∈ is called an eigenvalue of a pen�cil A – μB: �n → �n if there is a nontrivial vector x* ∈�n such that

By analogy with the definition of the rank of a pencil,we say that μ = ∞ is an eigenvalue of the pencil if thereis a nontrivial vector x* ∈ �n such that Bx* = 0. The setof all eigenvalues of A – μB: �n → �n (i.e., the spec�trum) is denoted by σ(A, B).

This definition implies that the spectrum of a sin�gular pencil A – μB: �n → �n coincides with the whole

complex plane: σ(A, B) = .

Among the points of the spectrum σ(A, B), we dis�tinguish numbers μ*, referred to as regular eigenvalueof A – μB: �n → �n, such that

(3)

L μ( ) A μB: �n– �

n.→=

nrank A μB–( ) rank A μB–( ).μ �∈

max=

1μ��A B–⎝ ⎠

⎛ ⎞

�

Ax* μ*Bx*.=

�

rank A μ*B–( ) nrank A μB–( ).<

The set of all regular eigenvalues is called the regularspectrum of the pencil A – μB: �n → �n and is denotedby σR(A, B).

Let Mn denote the space of all linear operators from�n to �n. Consider the set of all pairs of matrices (A, B)that form a pencil of prescribed normal rank m:

Let (A, B) ∈ Sn. Then the regular pencil A – μB has neigenvalues μ1(A, B), μ2(A, B), …, μn(A, B) (countingmultiplicities). Each of the eigenvalues μj(A, B) is a

continuous functional from Sn to in the following

sense: if (Ak, Bk) ⊂ Sn and (Ak, Bk) → (A, B), thenμj(Ak, Bk) → μj(A, B). The continuity of the eigenvalueμ*(A, B) in the regular pencil is equivalent to the factthat ∀(A1, B1) ∈ Mn × Mn

(4)

Now we consider a perturbed pencil

(5)

such that (A, B) ∈ Sm and det(L(μ, ε)) 0. It will beshown below that, in contrast to a perturbed regularpencil, the limit as ε → 0 of an arbitrary eigenvalueμk(ε) = μk(A, B, A1, B1, ε) is a functional not only ofthe matrices A and B but also of A1 and B1:

(6)

This means that, in the general case, the eigenvalues ofa singular pencil are discontinuous functionals atpoints of Sm.

At the same time, if μ* is a regular eigenvalue of asingular pencil A – μB, then we can always (i.e., forany perturbations A1 and B1) find an eigenvalue μ*(A +εA1, B + εB1) such that (4) holds. In other words, anyregular eigenvalue of A – μB is continuous on Sm. Thisfollows from the fact that μ* ∈ σR(A, B) if and only ifμ – μ* is a common divisor of all the minors of orderm = nrank(A – μ*B) for the pencil A – μB: �n → �n.

Sm A B,( ) Mn∈ Mn nrank A μB–( )× m={ }.=

�

}k 1=∞

μ* A εA1+ B εB1+,( )ε 0→lim μ* A B,( ).=

L μ ε,( ) A εA1 μ B εB1+( ): �n–+ �

n→=

≡

μ* ε( )ε 0→lim μ* A εA1+ B εB1+,( )

ε 0→lim=

= μ* A B A1 B1, , ,( ).

On Quasiregular Spectrum of Matrix PencilsN. F. Valeev

Presented by Academician V.A. Sadovnichii March 4, 2013

Received March 21, 2013

DOI: 10.1134/S1064562413050190

Institute of Mathematics and Computing Center, Ufa Scientific Center, Russian Academy of Sciences, ul. Chernyshevskogo 112, Ufa, 450008 Bashkortostan, Russiae�mail: valeevn@mail.ru, valeevnf@yandex.ru

MATHEMATICS

546

DOKLADY MATHEMATICS Vol. 88 No. 2 2013

VALEEV

Thus, under the condition det(L(μ, ε)) 0, per�turbed pencil (5) has eigenvalues whose limits as ε → 0vary depending on the “direction” of the perturbation(A1, B1) ∈ Mn × Mn and eigenvalues whose limits areinvariant under the perturbation matrices.

Accordingly, the problem arises of describing theset (denoted hereafter by σQ(A, B)) consisting of thelimiting values of eigenvalues of the pencil L(μ, ε) thatsatisfy condition (4) irrespective of the perturbationmatrices A1 and B1.

The above�described properties of matrix pencilsarise and play an important role in the construction ofperturbation�stable solutions to multiparameterinverse eigenvalue problems (see [7, 6]) and in the the�ory of multiparameter eigenvalue problems (see [4, 5,8]) in the so�called “uncertain” case.

In connection with what was said above, we give thefollowing definition.

Definition. The quasiregular spectrum of a pencilL(μ) = A – μB: �n → �n is the set

Points of the quasiregular spectrum are called quasi�regular eigenvalues.

Below, we derive formulas for computing quasireg�ular eigenvalues of singular pencils and describe someproperties of singular pencils and their perturbationsclosely related to the quasiregular spectrum. Accord�ingly, the following two assumptions are made (with�out loss of generality):

(7)

If rankB < nrank(A – μB), then a pencil satisfyingconditions (7) can be obtained by applying a linearfractional transformation of μ.

Indeed, it follows from (2) that there exists μ* ∈ �such that rank(A + μ*B) = m. Then

Now, setting

we pass to the equivalent pencil L0(s) = A0 – sB0,where rankB0 = m. The second condition in (7) can beachieved by passing to the strictly equivalent pencilL∗(μ) = UL(μ)V, where U, V: �n → �n are the unitarymatrices involved in a singular value decompositionof B; i.e., B∗ = BUV.

Under these transformations, the quasiregularspectra of the pencils L0(s) and L(μ) are related by the

linear fractional transformation s = .

2. First, we formulate the properties of singularpencils given by (1) and (7), which are then used toobtain formulas for computing quasiregular eigenval�

≡

σQ A B,( ) σ A εA1+ B εB1+,( )ε 0→lim{ }.

A1 B1, Mn∈

∩=

rankB nrank A μB–( ) m, B* B 0.≥= = =

A μB– μ* μ–μ*

������������� A μμ μ*–������������� A μ*B–( )– .=

s μμ μ*–�������������, B0 A μ*B, A0– A,= = =

μμ μ*–�������������

ues. Note that, in the author’s view, these properties ofpencils are new and of interest on their own.

Let P be a self�adjoint projector onto the subspace

V2 = kerB, V1 = . Then, in a suitable basis, the pen�cil L(μ) can be represented as

(8)

where A11 = (I – P)A(I – P): V1 → V1, A12 = (I – P)A P:V2 → V1, A21 = PA(I – P): V1 → V2, and A22 = PAP:V2 → V2. Here, L11(μ) = A11 – μB: V1 → V1 is a regularpencil, while A11 – μB and A22 are square matrices ofsizes m × m and (n – m) × (n – m), respectively.

Obviously, the condition B = nrank(A – μB) = mleads to certain relations between the blocks inmatrix (8). More specifically, the following result istrue.

Theorem 1. Let conditions (7) hold. Then, for anyμ ∈ �, representation (8) of the pencil L(μ) is such that

(9)

Combining identity (9) with the representation ofeigenprojectors of regular pencils in terms of contourintegrals of the resolvent yields the following propertyof eigenvalues and corresponding right and left eigen�vectors of A11 – μB.

Theorem 2. Let μ0 ∈ � be an arbitrary eigenvalue ofthe pencil L11(μ) = A11 – μB: V1 → V1, and let A11x0 =

μ0Bx0 and y0 = 0By0.

Then either A21x0 = 0 or y0 = 0.

Under the conditions of this theorem, it may hap�pen that A21x0 = 0 and y0 = 0. In this case, we canshow that the corresponding eigenvalue μ0 ∈ � of thepencil A11 – μB: V1 → V1 is a regular eigenvalue of L(μ).

The next result in this section is stated for pencils ofthe form

(10)

where nrankC(μ) = nrank(C11 – μI1) = m < n, V2 =

kerI1, and V1 = .

The pencils L(μ) and C(μ) are strictly equivalent,because

C11 = B–1A11, C12 = B–1A12, C21 = A21, and I1 and I2 arethe identity operators in V1 and V2, respectively.

Theorem 3. Let the normal rank of a pencil C(μ) ofform (10) be m, where m < n. Then the polynomial

(11)

V2⊥

L μ( ) A11 μB– A12

A21 A22⎝ ⎠⎜ ⎟⎛ ⎞

,=

A22 A12 A11 μB–( ) 1– A21– 0.≡

A11* μ

A12*

A12*

C μ( ) C11 μI1– C12

C21 0⎝ ⎠⎜ ⎟⎛ ⎞

: �n

�n,→=

V 2⊥

C μ( ) B 1– 0

0 I2⎝ ⎠⎜ ⎟⎛ ⎞ A11 μB– A12

A21 0⎝ ⎠⎜ ⎟⎛ ⎞

,=

q μ( ) det C11 – μI1( )=

DOKLADY MATHEMATICS Vol. 88 No. 2 2013

ON QUASIREGULAR SPECTRUM OF MATRIX PENCILS 547

is the greatest common divisor of all mth order minors ofthe pencil [C(μ)]2.

In other words, the eigenvalues of A11 – μB coin�cide with the regular eigenvalues of [C(μ)]2.

The last assertion can be illustrated using a simpleexample. Consider the pencil

For all μ ∈ , it is easy to see that rankC(μ) ≡ 3. Thesquare of C(μ) is written as

Now rank[C(μ)]2 = 3 for all μ ≠ μk, but rank[C(μk)]2 = 2for k = 1, 2, 3.

Note that the greatest common divisor of allmth�order minors of the matrix C(μ) is 1, while thecorresponding minors of [C(μ)]2 are divided by (μ1 –μ)(μ2 – μ)(μ3 – μ) = det[C(μ)].

3. In view of Theorem 3, the quasiregular spectrumof pencil (1) can be analyzed using the Smith normalform (see, e.g., [2, 3]). Applying the perturbation the�ory to the Smith normal form of the polynomial pencil[L(μ)]2, we obtain the main result of this paper.

Theorem 4. Suppose that L(μ) = A – μB: �n → �n isa pencil satisfying conditions (7), and let P be the orthog�onal projector onto the subspace kerB. Then the quasi�regular spectrum σQ(A, B) of L(μ) coincides with thezero set of the polynomial

(12)

Theorems 2–4 imply that the regular spectrumσR(A, B) of pencil (1) is a subset of the quasiregularpencil.

In some cases, they may coincide. Specifically, thefollowing result is true.

Theorem 5. If L(μ) = A – μB: �n → �n is a self�adjoint pencil for real μ, then σQ(A, B) = σR(A, B).

The regular spectrum of a pencil L(μ) = A – μB:�n → �n is stable with respect to perturbations in thefollowing sense. Let μ0 be a regular eigenvalue of L(μ) =

A – μB. Then, for any perturbed pencil L(μ, ε) = A +εA1 – μ(B + εB1), there exists an eigenvalue μ(ε) and acorresponding eigenvector u(ε) such that μ(0) = μ0,(A – μ0B)u(0) = 0, and Bu(0) ≠ 0.

Below is a similar assertion for the quasiregularspectrum.

Theorem 6. Suppose that a pencil L(μ, ε) = A + εA1 –μ(B + εB1): �n → �n satisfies the conditionsdet[L(μ, ε)] 0 and μ0 ∈ σQ(A, B).

Then there is an eigenvalue μ(ε) of L(μ, ε) and cor�responding right and left eigenvectors u(μ) and v(ε) ofunit length (i.e., L(μ, ε)u(ε) = 0 and [L(μ, ε)]*v(ε) = 0)such that

(a) μ(0) = μ0 and

(b) either ||Bu(0)|| ≠ 0 or ||B*v(0)|| ≠ 0.

To conclude, we note that the quasiregular spec�trum discussed in this paper is a natural extension anddevelopment of some approaches in the theory of mul�tiparameter eigenvalue problems. For example, regu�lar solutions of multiparameter eigenvalue problemsand methods for constructing such solutions areaddressed in [5, 7–9].

The quasiregular spectrum defined above is easy toextend to polynomial pencils and the multiparameterpencils considered in [5, 7–9]. In turn, this makes itpossible to expand the concept of stable solutions ofmultiparameter inverse eigenvalue problems, whichare important in various applications.

REFERENCES

1. F. R. Gantmacher, The Theory of Matrices (Chelsea,New York, 1959; Fizmatlit, Moscow, 2004).

2. P. R. Halmos, Finite Dimensional Vector Spaces (Princ�eton Univ. Press, Princeton, 1947; Fizmatgiz, Moscow,1963).

3. M. Marcus and H. Minc, A Survey of Matrix Theory andMatrix Inequalities (Allyn and Bacon, Boston, 1964;Nauka, Moscow, 1972).

4. F. M. Atkinson, Multiparameter Eigenvalue Problems,Vol. 1: Matrices (McGraw�Hill, New York, 1972).

5. M. E. Hochstenbach, A. Muhic, and B. Plestenjak,Linear Algebra Appl. 436, 2725–2743 (2012).

6. V. A. Sadovnichii, Ya. T. Sultanaev, and N. F. Valeev,Dokl. Math. 79, 390–393 (2009).

7. N. F. Valeev, Math. Notes 85, 890–893 (2009).

8. A. Muhic and B. Plestenjak, Electron. J. Linear Algebra18, 420–437 (2009).

9. V. N. Kublanovskaya and V. B. Khazanov, J. Math. Sci.(New York) 141, 1668–1677 (2007).

10. V. N. Kublanovskaya, J. Math. Sci. (New York) 182,823–829 (2012).

Translated by I. Ruzanova

C μ( )

μ1 μ– 0 0 0 0

0 μ2 μ– 0 1 0

0 0 μ3 μ– 0 1

1 0 0 0 0

0 0 0 0 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

�

C μ( )

=

μ1 μ–( )2 0 0 0 0

1 μ2 μ–( )2 0 μ2 μ– 0

0 0 μ3 μ–( )2 0 μ3 μ–

μ1 μ– 0 0 0 0

0 0 0 0 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.

q μ( ) det I P–( ) A μB–( ) I P–( )[ ].=

≡