Hyperbolic Schur Decomposition

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Linear Algebra and its Applications 440 (2014) 90110

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

The hyperbolic Schur decomposition

Vedran ego a,b,a Faculty of Science, University of Zagreb, Croatiab

School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 December 2012Accepted 22 October 2013Available online 14 November 2013Submitted by F. Dopico

MSC:15A63

46C2065F25

Keywords:Indefinite scalar productsHyperbolic scalar productsSchur decomposition

Jordan decompositionQuasitriangularizationQuasidiagonalizationStructured matrices

We propose a hyperbolic counterpart of the Schur decomposition,with the emphasis on the preservation of structures related tosome given hyperbolic scalar product. We give results regardingthe existence of such a decomposition and research the propertiesof its block triangular factor for various structured matrices.

2013 Elsevier Inc. All rights reserved.

1. Introduction

The Schur decomposition A=U T U , sometimes also called Schurs unitary triangularization, isa unitary similarity between any given square matrix A Cnn and some upper triangular matrixT Cnn . Such a decomposition has a structured form for various structured matrices, i.e., T is diag-onal if and only if A is normal, real diagonal if and only if A is Hermitian, positive (nonnegative) realdiagonal if and only if A is positive (semi)definite and so on.

Furthermore, the Schur decomposition can be computed in a numerically stable way, making ita good choice for calculating the eigenvalues of A (which are the diagonal elements of T) as well as

* Correspondence to: School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK.E-mail address:[email protected].

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V. ego / Linear Algebra and its Applications 440 (2014) 90110 91

the various matrix functions (for more details, see [11]). Its structure preserving property allows tosave time and memory when working with structured matrices. For example, computing the value ofsome function of a Hermitian matrix is reduced to working with a diagonal matrix, which involvesonly evaluation of the diagonal elements.

Unitary matrices are very useful when working with the traditional Euclidean scalar productx,y =yx, as their columns form an orthonormal basis ofCn . However, many applications requirea nonstandard scalar product which is usually defined by[x,y]J=yJ x, where Jis some nonsingu-lar matrix, and many of these applications consider Hermitian or skew-Hermitian J. The hyperbolicscalar product defined by a signature matrix J=diag(j1, . . . ,jn), where jk {1, 1} for all k, arisesfrequently in applications. It is used, for example, in the theory of relativity and in the research of thepolarized light. More on the applications of such products can be found in [10,13,14,17].

The Euclidean matrix decompositions have some nice structure preserving properties even in non-standard scalar products, as shown by Mackey, Mackey and Tisseur [16],but it is often worth lookinginto versions of such decompositions that respect the structures related with the given scalar prod-uct. There is plenty of research on the subject, i.e., hyperbolic SVD [17,24], J1J2-SVD[9], two-sided

hyperbolic SVD[20],hyperbolic CS decomposition[8,10]and indefinite QR factorization[19].There are many advantages of using decompositions related to some specific, nonstandard scalarproduct, as such decompositions preserve structures related to a given scalar product. They can sim-plify calculation and provide a better insight into the structures of such structured matrices.

In this paper we investigate the existence of a decomposition which would resemble the traditionalSchur decomposition, but with respect to the given hyperbolic scalar product. In other words, oursimilarity matrix should be unitary-like (orthonormal, to be more precise) with respect to that scalarproduct.

As we shall see, a hyperbolic Schur decomposition can be constructed, but not for all squarematrices. Furthermore, we will have to relax conditions on both U and T. The matrix Uwill be hy-perexchange (a column-permutation of the matrix unitary with respect to J). The matrix Twill have

to be block upper triangular with diagonal blocks of order 1 and 2. Both of these changes are quiteusual in hyperbolic scalar products. For example, they appear in the traditional QR vs. the hyperbolicQR factorizations[19].

Some work on the hyperbolic Schur decomposition was done by Ammar, Mehl and Mehrmann [1,Theorem 8], but with somewhat different focus. They have assumed to have a partitioned J=Ip (Iq), in the paper denoted as p,q , for which they have observed a Schur-like similaritythrough unitary factors (without permuting J), producing more complex triangular factors. Also, theirdecomposition is applicable only to the set of J-unitary matrices, in the paper denoted as the Liegroup Op,q .

In the symplectic scalar product spaces, Schur-like decomposition was researched by Lin,Mehrmann and Xu[15],by Ammar, Mehl and Mehrmann[1], and by Xu[22,23].

In Section2, we provide a brief overview of the definitions, properties and other results relatingto the hyperbolic scalar products that will be used later. In Section3,the definition and the construc-tion of the hyperbolic Schur decomposition are presented. We also provide sufficient requirementsfor its existence and examples showing why such a decomposition does not exist for all matrices.In Section 4 we observe various properties of the proposed decomposition. We finalize the resultsby providing the necessary and the sufficient conditions for the existence of the hyperbolic Schurdecomposition of J-Hermitian matrices in Section5.

The notation used is fairly standard. The capital letters refer to matrices and their blocks, elementsare denoted by the appropriate lowercase letter with two subscript indices, while lowercase letterswith a single subscript index represent vectors (including matrix columns). By J=diag(1) we de-note a diagonal signature matrix defining the hyperbolic scalar product, while P and Pk (for some

indicesk) denote permutation matrices. We use

Sn:= [i,n+1j] = 1. . .

1

= S1n


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92 V. ego / Linear Algebra and its Applications 440 (2014) 90110

for the standard involutory permutation (see [6, Example 2.1.1]), J for a Jordan matrix and Jk()for a single Jordan block of order k associated with the eigenvalue . Vector ek denotes k-th columnof the identity matrix and denotes the Kronecker product. The symbol is used to describe adiagonal concatenation of matrices, i.e., AB is a block diagonal matrix with the diagonal blocks Aand B .Also a standard notation, but somewhat incorrect in terms of the indefinite scalar products, is|v| := |[v, v]|. This is used as the norm of vector v induced by the scalar product [,], but oneshould keep in mind that it doesnt have the usual properties of the norm (definiteness and thetriangle inequality do not hold), but is used nevertheless due to its relation with the scalar prod-uct.

2. The hyperbolic scalar products

As mentioned in the introduction, an indefinite scalar productis defined by a nonsingular Hermitianindefinite matrix J Cnn as[x,y]J= yJ x. When J is known from the context, we simply write

[x,y

]instead of

[x,y

]J.

When J is a signature matrix, i.e., J=diag(1):=diag(j11,j22, . . . ,jnn), where jkk {1, 1} forallk , the scalar product is referred to as hyperbolicand takes the form

[x,y]J=yJ x=n

i=1jiixiyi.

Throughout this paper we assume that all considered scalar products are hyperbolic, unless statedotherwise.

Indefinite scalar products have another important property which, unfortunately, causes a majorproblem with the construction of the decomposition. A vector v=0 is said to be J-degenerate if

[v, v

] =0; otherwise, we say that it is J-nondegenerate. Degenerate vectors are sometimes also called

J -neutral.If[v, v]0.

When Jis known from the context, we simply say that the vector is degenerate,nondegenerate,neutral,negativeor positive.

We extend this notion to matrices as well: a matrix A is J-degenerate if rankAJ A


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We say that U is J-hyperexchange ifUJ U= PJ P for some permutation P. Although the termhyperexchange is quite common, we often refer to such matrices as J-orthonormal, to emphasizethat their columns are J-orthonormal vectors. In other words, if the columns ofUare denoted as ui ,then

[ui , ui] = 1, [ui, uj] =0, for alli,j.More on the definitions and properties related to hyperbolic (and, more generally, indefinite) scalar

products can be found in[6].Throughout the paper, we often consider the diagonal blocks of a given matrix A. In order to keep

the relation with the given hyperbolic scalar product induced by some J=diag(1), we introducethe term the corresponding part of J. Let J=diag(j1, . . . ,jn), jk {1, 1}, define a hyperbolic scalarproduct and let A= [Ai j] Cnn be a blockmatrix with elements apq partitioned in the blocks A i jin a way that each diagonal block Akk is of order 1 or 2. Observing the block Akk for a given k, thecorresponding part of J, here denoted as J , is defined as J= [jp] if Akk= [app] is of order 1 and asJ

=diag(jp ,jp

+1)if

Akk=

app ap,p+1ap+1,p ap+1,p+1

is of order 2.

3. Definition and existence of the hyperbolic Schur decomposition

In this section we present the definition and the main results regarding the hyperbolic Schurdecomposition of a matrix ACnn .

Like other hyperbolic generalizations of the Euclidean decompositions, this one also has a hy-perexchange matrix instead of a unitary one, as well as the block structured factor instead of atriangular/diagonal factor of the Schur decomposition. Block upper triangular matrices with the diag-onal blocks of order 1 and 2 are usually referred to as quasitriangular(see[18, Section 3.3]). Similarly,ifTis block diagonal with the diagonal blocks of order 1 and 2, we refer to it as quasidiagonal.

Before providing a formal definition, let us first show obstacles which will explain why thesechanges are necessary.

It is a well known fact that the first column u1 of the similarity matrix Uin any similarity trian-gularization A=U T U1 is an eigenvector ofA. Also, it is easy to construct a nonsingular matrix withall degenerate columns (see[21, Example 3.1]), which cannot be J-normalized. This means that thereare some matrices (even diagonalizable ones!) that are not unitarily triangularizable with respect tothe scalar product induced by J. However, allowing the blocks of T to be of order 2, we relax thatcondition, as shown inExample 3.1.

For better clarity, we say that a block Xj j of a matrix X is irreducible if it cannot be split intosmaller blocks, without losing the block structure of the matrix. IfX=ki=1Xii is block diagonal, wesay that Xj j is irreducible if it cannot be split as Xj j=Y1 Y2 for some square blocks Y1 andY2. Fora block triangular X, we say that Xj j is irreducible if it cannot be split as Xj j=

Y11 Y12

Y22

, where Y11

and Y22 are square blocks. Since we mostly deal with blocks of order 2, the irreducible blocks willbe those of order 1 and those of order 2 with one of the nondiagonal elements (in case X is blockdiagonal) or the bottom left element (in case Xis a block triangular) being nonzero.

The described notation of irreducible blocks is just a descriptive way of saying that we cannotsplit Xinto smaller blocks while preserving its block diagonal or block triangular structure. It is usedto simplify some of the statements and proofs by reducing the number of observed cases without

losing generality. See[21, Example 3.2]for further clarification of the concept of irreducible blocks.Unfortunately, allowing the triangular factor to have blocks is not enough. We also need permu-

tationsof the decomposing matrix and the scalar product generator J. The following example showswhy this is needed, simultaneously illustrating a general approach to proving that some matrices donot have a hyperbolic Schur decomposition.


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Example 3.1.Let J=diag(1, 1, 1, 1)and let A=SJS1, where J=diag(1, 2, 3, 4)and

S=

15

31

17

151

195

71

257

2552

5

34

17

158

195

716

257

2554

5

316

17

1564

195

7256

257

2558

5

364

17

15512

195

74096

257

255

.

We shall now show that no J-unitary V and quasitriangular Texist such that A=V T V1 . Since

SJ S=

1 0.994393 0.970230 0.949852

0.994393 1 0.993828 0.9850750.970230 0.993828 1 0.9981510.949852 0.985075 0.998151 1

(2)to 6 significant digits, all eigenvectors of A (i.e., columns of S) are normalized negative vectors. Fur-thermore, if we denote them by si , then1


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But, since

(A t22I)(A t11I)=(A t11I)(A t22I),v1 and v2 are both (linearly independent) eigenvectors of A2, with the same eigenvalue .Since A and A2 are diagonalizable, every eigenvector of A is also an eigenvector of A2. Moreover,since the eigenvalues of A are distinct (they are 1, 2, 3 and 4), A2 has at most one eigenvalue ofmultiplicity 2 (this is easily seen from its Jordan decomposition). In other words, its eigenspaceshave dimensions at most 2, so v1 and v2 are linear combinations ofsi and sj for some i= j.Let v1=1si+1sj and v2=2si+2sj . Obviously, 1, 1=0 (or v1 would be an eigenvectorof A, hence covered by the case 1). Then, from J=VJ V, we get

1= j11= [v1, v1] = |1|2[si , si] + |1|2[sj , sj] + 2 Re

11[si, sj]

= |1|2 + |1|2 2[si, sj] Re(11). (5)From(2) we see that for i

=k,

1. In[6], J, X and Sare denoted as H, T1 and J, respectively.

The decomposition(13)is referred to as the canonical form of the pair(A, J) and has many appli-cations. See[6, Chapter 5]for the applications in the indefinite scalar product spaces and [15]for itssymplectic version and the application in the symplectic scalar product spaces.

The set of signs{1, . . . , }is called the sign characteristicof the pair (A, J). In[15], this notion isadapted to the symplectic scalar product spaces, losing the property that k=1 for k> . For moredetails, see[15, Section 3.2].

Apart from the applications to the nonstandard scalar product spaces, the sign characteristic plays

a significant role in the research of the selfadjoint matrix polynomials [5, Chapter 12].Throughout this paper, the Jordan normal form of a matrix plays a crucial role in the existence of

the hyperbolic Schur decomposition. To better describe it, we associate with each Jordan block Jd()the term partial multiplicity, which is the size d of that Jordan block. Obviously, each eigenvalue hasas many partial multiplicities as it has associated Jordan blocks and the largest partial multiplicity of


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each eigenvalue of a matrix A is equal to the size of its largest Jordan block or, equivalently, tomax{dN: (x)d|A (x)}, where A (x)is the minimal polynomial ofA.

We are now ready to present and prove the necessary and sufficient conditions for the existenceof the hyperbolic Shur decomposition of a J-Hermitian matrix.

Theorem 5.1(Hyperbolic Schur decomposition of a J-Hermitian matrix). Let J=diag(1)and let ACnnbe a J -Hermitian matrix of the same order. Then A has a hyperbolic Schur decomposition with respect to Jif and only if its real eigenvalues have partial multiplicities at most2and its nonreal eigenvalues have partialmultiplicities at most1.

Proof. Let us first show that if such a decomposition of A exists, then the real eigenvalues of A havepartial multiplicities at most 2 and its nonreal eigenvalues have multiplicities at most 1.

Let A=U T U1 , where U is J-orthonormal, and letJ:=UJ U. ByProposition 4.3, T is quasidi-agonal and

J-Hermitian. This means that we can write

T= Tk, J=kJk,

where each Tk is irreducible (i.e., either of order 1 or nondiagonal of order 2) andJk has the sameorder as Tk . Furthermore, each Tk isJk-Hermitian. Trivially, this means that all blocks Tk of order 1are real.

Notice that if Tk of order 2 has nonreal eigenvalues (k)1 and

(k)2 , then they form a complex

conjugate pair, i.e., (k)1 =(k)2 , due to[16, Theorem 7.6],so their partial multiplicity must be 1. If they

are real, their partial multiplicity is 1 or 2, depending on the diagonalizability ofTk . Since A is similarto Tk , it has the same Jordan structure, i.e., the same eigenvalues with the same partial multiplicities.

Let us now show that any J-Hermitian matrix A with the described eigenvalues partial multiplici-ties has a hyperbolic Schur decomposition with respect to J. We observe the canonical form of(A, J)from(13).By the assumption, all Jk are of an order at most 2.

Let k be such that Jk is of order 2. Then Sk=

11

, which is congruent to diag(1, 1). By the

Sylvester law of inertia, there exists Yk such that YkSkYk=diag(1, 1). For all k such that Jk is of

order 1 we define Yk=1, so YkSYk=I1 for such k . Furthermore, we define

Y:=

k

Yk, U:=X Y, T:=Y1JY. (14)

Note that Thas the same block structure as J, so it is quasidiagonal, and

YSY= k Ykk kSkk Yk=k k YkSkYk=: J, (15)where J=diag(j1, . . . ,jn)for some j1, . . . ,jn {1, 1}. Using(13),(14)and(15),we see that

A=XJX1 =U Y1JY U1 =U T U1,UJ U=(X Y)J(X Y)=YXJ X Y=YSY= J,

which proves that A is J-orthonormality similar to some quasidiagonal T, hence A has a hyperbolicSchur decomposition with respect to J.

6. Conclusion

In this paper we have introduced the hyperbolic Schur decomposition of a square matrix with re-spect to the scalar product induced by J=diag(1). We have shown that all diagonalizable matriceshave such a decomposition, which means that the set of matrices that dont have a hyperbolic Schurdecomposition is a subset of the set of nondiagonalizable matrices, which is a set of measure zero


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(this follows from[12, Section 2.4.7]). We have also given examples of matrices that do not have sucha decomposition.

By its design, the hyperbolic Schur decomposition preserves structures of the structured matrices,albeit with respect to a somewhat changed (symmetrically permuted) J, denoted by

J, which is a

common property of hyperbolic decompositions. We have analyzed the properties of such matrices,as well as the properties of atomic (indecomposable) blocks on the diagonal of the quasitriangularfactor T.

InExample 3.8 we have shown that there exist J-Hermitian (and, therefore, J-normal) matricesthat do not have a hyperbolic Schur decomposition, even in spaces as simple as the Minkowski space.InExample 3.9we have provided an example of a J-unitary matrix for which there is no hyperbolicSchur decomposition, also showing that not all hyperexchange matrices have such a decomposition(since J-unitary matrices are a special case).

In Section5,we have given sufficient and necessary conditions under which a J-Hermitian matrixhas a hyperbolic Schur decomposition.

We have shown that those structured matrices that do have a hyperbolic Schur decomposition also

have desirable properties. The only exemption from this rule are general J-normal matrices (whichare not nicer, i.e., neither J-Hermitian nor hyperexchange) for which block triangularity does notimply block diagonality. It remains to be researched if the block triangular ones can always be blockdiagonalized via J-orthonormal similarities (i.e., by a hyperbolic Schur decomposition), as alwayshappens in the Euclidean case. Interestingly enough, all important subclasses of J-normal matricesmaintain the block diagonality of the factor T.

Another subject that remains to be researched is the algorithm to calculate such a decomposition.The first thing that might come to mind here is the QR algorithm used for the Euclidean Schur de-composition. However, this would not work, at least not as directly as one might hope. Firstly, everynonsingular matrix has a hyperbolic QR factorization, as shown by [19, Theorem 5.3]. But, as shownin the examples in this paper, not all such matrices have a hyperbolic Schur decomposition (seeEx-ample 3.5 for

=0, Example 3.6 for

1,

2=0 andExample 3.9). This means that any algorithm

employing the hyperbolic QR factorization to calculate the hyperbolic Schur decomposition will di-verge for such matrices. Secondly, many singular matrices have a hyperbolic Schur decomposition,while it is unclear if they also have a hyperbolic QR factorization, since[19, Theorem 5.3]covers onlythe matrices A such that AJ A is of full rank. This means that it may be possible for some matricesto have a hyperbolic Schur decomposition which is uncomputable via the hyperbolic QR factorization.

Acknowledgements

Most of the work on this paper was done at the School of Mathematics, University of Manchester,where I was invited as a research visitor by Franoise Tisseur, whom I thank dearly for the opportunityand a great working experience as well as many suggestions that helped me with this work. The paperwas also proofread by Nataa Strabic whom I thank for all the suggestions that have considerablyimproved the paper. I would also like to thank the anonymous referee at the LAA, who made aspecial impact on this paper and without whose suggestions Section5 would not exist.

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Hyperbolic Schur Decomposition

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