NAG Library Chapter Introduction

F02 – Eigenvalues and Eigenvectors

The majority of routines for these problems can be found in Chapter F08 which contains software derived from LAPACK (see Anderson et al. (1999)). However, you should read the introduction to this chapter before turning to Chapter F08, especially if you are a new user. Chapter F12 contains routines for large sparse eigenvalue problems, although one such routine is also available in this chapter.

Chapters F02 and F08 contain Black Box (or Driver) routines that enable many problems to be solved by a call to a single routine, and the decision trees in Section 4 direct you to the most appropriate routines in Chapters F02 and F08. The Chapter F02 routines call routines in Chapters F07 and F08 wherever possible to perform the computations, and there are pointers in Section 4 to the relevant decision trees in Chapter F08.

Here we describe the different types of problem which can be tackled by the routines in this chapter, and give a brief outline of the methods used to solve them. If you have one specific type of problem to solve, you need only read the relevant sub-section and then turn to Section 3. Consult a standard textbook for a more thorough discussion, for example Golub and Van Loan (1996) or Parlett (1998).

In each sub-section, we first describe the problem in terms of real matrices. The changes needed to adapt the discussion to complex matrices are usually simple and obvious: a matrix transpose such as QT must be replaced by its conjugate transpose QH; symmetric matrices must be replaced by Hermitian matrices, and orthogonal matrices by unitary matrices. Any additional changes are noted at the end of the sub-section.

Equation (2) can be rewritten This is known as the eigen-decomposition or spectral factorization of A.

Eigenvalues of a real symmetric matrix are well-conditioned, that is, they are not unduly sensitive to perturbations in the original matrix A. The sensitivity of an eigenvector depends on how small the gap is between its eigenvalue and any other eigenvalue: the smaller the gap, the more sensitive the eigenvector. More details on the accuracy of computed eigenvalues and eigenvectors are given in the routine documents, and in the F08 Chapter Introduction.

For dense or band matrices, the computation of eigenvalues and eigenvectors proceeds in the following stages:

1. A is reduced to a symmetric tridiagonal matrix T by an orthogonal similarity transformation: A=QTQT, where Q is orthogonal. (A tridiagonal matrix is zero except for the main diagonal and the first subdiagonal and superdiagonal on either side.) T has the same eigenvalues as A and is easier to handle.
2. Eigenvalues and eigenvectors of T are computed as required. If all eigenvalues (and optionally eigenvectors) are required, they are computed by the QR algorithm, which effectively factorizes T as T=SΛST, where S is orthogonal. If only selected eigenvalues are required, they are computed by bisection, and if selected eigenvectors are required, they are computed by inverse iteration. If s is an eigenvector of T, then Qs is an eigenvector of A.

All the above remarks also apply – with the obvious changes – to the case when A is a complex Hermitian matrix. The eigenvectors are complex, but the eigenvalues are all real, and so is the tridiagonal matrix T.

If A is large and sparse, the methods just described would be very wasteful in both storage and computing time, and therefore an alternative algorithm, known as subspace iteration, is provided (for real problems only) to find a (usually small) subset of the eigenvalues and their corresponding eigenvectors. Chapter F12 contains routines based on the Lanczos method for real symmetric large sparse eigenvalue problems, and these routines are usually more efficient than subspace iteration.

A real nonsymmetric matrix A may have complex eigenvalues, occurring as complex conjugate pairs. If x is an eigenvector corresponding to a complex eigenvalue λ, then the complex conjugate vector x- is the eigenvector corresponding to the complex conjugate eigenvalue λ-. Note that the vector x defined in equation (1) is sometimes called a right eigenvector; a left eigenvector y is defined by Routines in this chapter only compute right eigenvectors (the usual requirement), but routines in Chapter F08 can compute left or right eigenvectors or both.

The eigenvalue problem can be solved via the Schur factorization of A, defined as where Z is an orthogonal matrix and T is a real upper quasi-triangular matrix, with the same eigenvalues as A. T is called the Schur form of A. If all the eigenvalues of A are real, then T is upper triangular, and its diagonal elements are the eigenvalues of A. If A has complex conjugate pairs of eigenvalues, then T has 2 by 2 diagonal blocks, whose eigenvalues are the complex conjugate pairs of eigenvalues of A. (The structure of T is simpler if the matrices are complex – see below.)

For example, the following matrix is in quasi-triangular form and has eigenvalues 1, 2±i, and 3. (The elements indicated by ‘*’ may take any values.)

The columns of Z are called the Schur vectors. For each k1≤k≤n, the first k columns of Z form an orthonormal basis for the invariant subspace corresponding to the first k eigenvalues on the diagonal of T. (An invariant subspace (for A) is a subspace S such that for any vector v in S, Av is also in S.) Because this basis is orthonormal, it is preferable in many applications to compute Schur vectors rather than eigenvectors. It is possible to order the Schur factorization so that any desired set of k eigenvalues occupy the k leading positions on the diagonal of T, and routines for this purpose are provided in Chapter F08.

Note that if A is symmetric, the Schur vectors are the same as the eigenvectors, but if A is nonsymmetric, they are distinct, and the Schur vectors, being orthonormal, are often more satisfactory to work with in numerical computation.

Eigenvalues and eigenvectors of a nonsymmetric matrix may be ill-conditioned, that is, sensitive to perturbations in A. Chapter F08 contains routines which compute or estimate the condition numbers of eigenvalues and eigenvectors, and the F08 Chapter Introduction gives more details about the error analysis of nonsymmetric eigenproblems. The accuracy with which eigenvalues and eigenvectors can be obtained is often improved by balancing a matrix. This is discussed further in Section 3.4.

Computation of eigenvalues, eigenvectors or the Schur factorization proceeds in the following stages:

1. A is reduced to an upper Hessenberg matrix H by an orthogonal similarity transformation: A=QHQT, where Q is orthogonal. (An upper Hessenberg matrix is zero below the first subdiagonal.) H has the same eigenvalues as A, and is easier to handle.
2. The upper Hessenberg matrix H is reduced to Schur form T by the QR algorithm, giving the Schur factorization H=STST. The eigenvalues of A are obtained from the diagonal blocks of T. The matrix Z of Schur vectors (if required) is computed as Z=QS.
3. After the eigenvalues have been found, eigenvectors may be computed, if required, in two different ways. Eigenvectors of H can be computed by inverse iteration, and then pre-multiplied by Q to give eigenvectors of A; this approach is usually preferred if only a few eigenvectors are required. Alternatively, eigenvectors of T can be computed by back-substitution, and pre-multiplied by Z to give eigenvectors of A.

All the above remarks also apply – with the obvious changes – to the case when A is a complex matrix. The eigenvalues are in general complex, so there is no need for special treatment of complex conjugate pairs, and the Schur form T is simply a complex upper triangular matrix.

As for the symmetric eigenvalue problem, if A and is large and sparse then it is generally preferable to use an alternative method. Chapter F12 provides routines based on Arnoldi's method for both real and complex matrices, intended to find a subset of the eigenvalues and vectors.

The singular value decomposition of A is closely related to the eigen-decompositions of the symmetric matrices ATA or AAT, because:

ATAvi=σi2vi  and  AATui=σi2ui.

However, these relationships are not recommended as a means of computing singular values or vectors unless A is sparse and routines from Chapter F12 are to be used.

If Uk, Vk denote the leading k columns of U and V respectively, and if Σk denotes the leading principal submatrix of Σ, then is the best rank-k approximation to A in both the 2-norm and the Frobenius norm.

Singular values are well-conditioned; that is, they are not unduly sensitive to perturbations in A. The sensitivity of a singular vector depends on how small the gap is between its singular value and any other singular value: the smaller the gap, the more sensitive the singular vector. More details on the accuracy of computed singular values and vectors are given in the routine documents and in the F08 Chapter Introduction.

The singular value decomposition is useful for the numerical determination of the rank of a matrix, and for solving linear least squares problems, especially when they are rank-deficient (or nearly so). See Chapter F04.

Computation of singular values and vectors proceeds in the following stages:

1. A is reduced to an upper bidiagonal matrix B by an orthogonal transformation A=U1BV1T, where U1 and V1 are orthogonal. (An upper bidiagonal matrix is zero except for the main diagonal and the first superdiagonal.) B has the same singular values as A, and is easier to handle.
2. The SVD of the bidiagonal matrix B is computed as B = U2 Σ V2T , where U2 and V2 are orthogonal and Σ is diagonal as described above. Then in the SVD of A, U=U1U2 and V=V1V2.

All the above remarks also apply – with the obvious changes – to the case when A is a complex matrix. The singular vectors are complex, but the singular values are real and non-negative, and the bidiagonal matrix B is also real.

By formulating the problems appropriately, real large sparse singular value problems may be solved using the symmetric eigenvalue routines in Chapter F12.

When B is nonsingular, equation (4) is mathematically equivalent to B-1Ax=λx, and when A is nonsingular, it is equivalent to A-1Bx=1/λx. Thus, in theory, if one of the matrices A or B is known to be nonsingular, the problem could be reduced to a standard eigenvalue problem.

However, for this reduction to be satisfactory from the point of view of numerical stability, it is necessary not only that B (or A) should be nonsingular, but that it should be well-conditioned with respect to inversion. The nearer B is to singularity, the more unsatisfactory B-1A will be as a vehicle for determining the required eigenvalues. Well-determined eigenvalues of the original problem (4) may be poorly determined even by the correctly rounded version of B-1A.

We consider first a special class of problems in which B is known to be nonsingular, and then return to the general case in the following sub-section.

If Λ is the diagonal matrix whose diagonal elements are the eigenvalues, and Z is the matrix whose columns are the eigenvectors, then

ZTAZ=Λ  and  ZTBZ=I.

To compute eigenvalues and eigenvectors, the problem can be reduced to a standard symmetric eigenvalue problem, using the Cholesky factorization of B as LLT or UTU (see Chapter F07). Note, however, that this reduction does implicitly involve the inversion of B, and hence this approach should not be used if B is ill-conditioned with respect to inversion.

For example, with B=LLT, we have Hence the eigenvalues of Az=λBz are those of Cy=λy, where C is the symmetric matrix C = L-1 AL-T  and y=LTz. The standard symmetric eigenproblem Cy=λy may be solved by the methods described in Section 2.1.1. The eigenvectors z of the original problem may be recovered by computing z = L-T y .

Most of the routines which solve this class of problems can also solve the closely related problems where again A and B are symmetric and B is positive definite. See the routine documents for details.

All the above remarks also apply – with the obvious changes – to the case when A and B are complex Hermitian matrices. Such problems are called Hermitian-definite. The eigenvectors are complex, but the eigenvalues are all real.

If A and B are large and sparse, reduction to an equivalent standard eigenproblem as described above would almost certainly result in a large dense matrix C, and hence would be very wasteful in both storage and computing time. The methods of subspace iteration and Lanczos type methods, mentioned in Section 2.1.1, can also be used for large sparse generalized symmetric-definite problems.

The generalized eigenvalue problem can be solved via the generalized Schur factorization of A and B:

A=QUZT,  B=QVZT

where Q and Z are orthogonal, V is upper triangular, and U is upper quasi-triangular (defined just as in Section 2.1.2).

If all the eigenvalues are real, then U is upper triangular; the eigenvalues are given by λi=uii/vii. If there are complex conjugate pairs of eigenvalues, then U has 2 by 2 diagonal blocks.

Eigenvalues and eigenvectors of a generalized nonsymmetric problem may be ill-conditioned; that is, sensitive to perturbations in A or B.

Particular care must be taken if, for some i, uii=vii=0, or in practical terms if uii and vii are both small; this means that the pencil is singular, or approximately so. Not only is the particular value λi undetermined, but also no reliance can be placed on any of the computed eigenvalues. See also the routine documents.

Computation of eigenvalues and eigenvectors proceeds in the following stages.

1. The pencil A-λB is reduced by an orthogonal transformation to a pencil H-λK in which H is upper Hessenberg and K is upper triangular: A = Q1 H Z1T  and B = Q1 K Z1T . The pencil H-λK has the same eigenvalues as A-λB, and is easier to handle.
2. The upper Hessenberg matrix H is reduced to upper quasi-triangular form, while K is maintained in upper triangular form, using the QZ algorithm. This gives the generalized Schur factorization: H=Q2UZ2 and K=Q2VZ2.
3. Eigenvectors of the pencil U-λV are computed (if required) by back-substitution, and pre-multiplied by Z1Z2 to give eigenvectors of A.

All the above remarks also apply – with the obvious changes – to the case when A and B are complex matrices. The eigenvalues are in general complex, so there is no need for special treatment of complex conjugate pairs, and the matrix U in the generalized Schur factorization is simply a complex upper triangular matrix.

As for the generalized symmetric-definite eigenvalue problem, if A and B are large and sparse then it is generally preferable to use an alternative method. Chapter F12 provides routines based on Arnoldi's method for both real and complex matrices, intended to find a subset of the eigenvalues and vectors.

Routines in the NAG Library for solving eigenvalue problems fall into two categories.

1. Black Box Routines: these are designed to solve a standard type of problem in a single call – for example, to compute all the eigenvalues and eigenvectors of a real symmetric matrix. You are recommended to use a black box routine if there is one to meet your needs; refer to the decision tree in Section 4.1 or the index in Section 5.
2. General Purpose Routines: these perform the computational subtasks which make up the separate stages of the overall task, as described in Section 2 – for example, reducing a real symmetric matrix to tridiagonal form. General purpose routines are to be found, for historical reasons, some in this chapter, a few in Chapter F01, but most in Chapter F08. If there is no black box routine that meets your needs, you will need to use one or more general purpose routines.

The decision trees in Section 4.2 list the combinations of general purpose routines which are needed to solve many common types of problem.

Sometimes a combination of a black box routine and one or more general purpose routines will be the most convenient way to solve your problem: the black box routine can be used to compute most of the results, and a general purpose routine can be used to perform a subsidiary computation, such as computing condition numbers of eigenvalues and eigenvectors.

Routines are provided in Chapter F08 for performing either or both of these preprocessing steps, and also for transforming computed eigenvectors or Schur vectors back to those of the original matrix.

The black box routines in this chapter which compute eigenvectors perform both forms of balancing.

Eigenvectors, as defined by equations (1) or (4), are not uniquely defined. If x is an eigenvector, then so is kx where k is any nonzero scalar. Eigenvectors computed by different algorithms, or on different computers, may appear to disagree completely, though in fact they differ only by a scalar factor (which may be complex). These differences should not be significant in any application in which the eigenvectors will be used, but they can arouse uncertainty about the correctness of computed results.

Even if eigenvectors x are normalized so that x2=1, this is not sufficient to fix them uniquely, since they can still be multiplied by a scalar factor k such that k=1. To counteract this inconvenience, most of the routines in this chapter, and in Chapter F08, normalize eigenvectors (and Schur vectors) so that x2=1 and the component of x with largest absolute value is real and positive. (There is still a possible indeterminacy if there are two components of equal largest absolute value – or in practice if they are very close – but this is rare.)

In symmetric problems the computed eigenvalues are sorted into ascending order, but in nonsymmetric problems the order in which the computed eigenvalues are returned is dependent on the detailed working of the algorithm and may be sensitive to rounding errors. The Schur form and Schur vectors depend on the ordering of the eigenvalues and this is another possible cause of non-uniqueness when they are computed. However, it must be stressed again that variations in the results from this cause should not be significant. (Routines in Chapter F08 can be used to transform the Schur form and Schur vectors so that the eigenvalues appear in any given order if this is important.)

In singular value problems, the left and right singular vectors u and v which correspond to a singular value σ cannot be normalized independently: if u is multiplied by a factor k such that k=1, then v must also be multiplied by k.

Non-uniqueness also occurs among eigenvectors which correspond to a multiple eigenvalue, or among singular vectors which correspond to a multiple singular value. In practice, this is more likely to be apparent as the extreme sensitivity of eigenvectors which correspond to a cluster of close eigenvalues (or of singular vectors which correspond to a cluster of close singular values).

The decision tree for this section is divided into three sub-trees.

• Tree 1 Eigenvalues and Eigenvectors of Real Matrices
• Tree 2 Eigenvalues and Eigenvectors of Complex Matrices
• Tree 3 Singular Values and Singular Vectors

Note: for the Chapter F08 routines there is generally a choice of simple and comprehensive routine. The comprehensive routines return additional information such as condition and/or error estimates.

Routines for large sparse eigenvalue problems are to be found in Chapter F12, see the F12 Chapter Introduction.

The decision tree for this section addressing dense problems, is divided into eight sub-trees:

• Tree 1 Real Symmetric Eigenvalue Problems in the F08 Chapter Introduction
• Tree 2 Real Generalized Symmetric-definite Eigenvalue Problems in the F08 Chapter Introduction
• Tree 3 Real Nonsymmetric Eigenvalue Problems in the F08 Chapter Introduction
• Tree 4 Real Generalized Nonsymmetric Eigenvalue Problems in the F08 Chapter Introduction
• Tree 5 Complex Hermitian Eigenvalue Problems in the F08 Chapter Introduction
• Tree 6 Complex Generalized Hermitian-definite Eigenvalue Problems in the F08 Chapter Introduction
• Tree 7 Complex non-Hermitian Eigenvalue Problems in the F08 Chapter Introduction
• Tree 8 Complex Generalized non-Hermitian Eigenvalue Problems in the F08 Chapter Introduction

As it is very unlikely that one of the routines in this section will be called on its own, the other routines required to solve a given problem are listed in the order in which they should be called.

See Section 4.2 in the F08 Chapter Introduction. For real sparse matrices where only selected singular values are required (possibly with their singular vectors), routines from Chapter F12 may be applied to the symmetric matrix ATA; see Section 9 in F12FBF.