Extremal Properties of Balanced Tri-Diagonal Matrices

If A is a square matrix with distinct eigenvalues and D a nonsingular matrix, then the angles between rowand column-eigenvectors of D~XAD differ from the corresponding quantities of A. Perturbation analysis of the eigenvalue problem motivates the minimization of functions of these angles over the set of diagonal similarity transforms ; two such functions which are of particular interest are the spectral and the Euclidean condition numbers of the eigenvector matrix X of D~lAD. It is shown that for a tri-diagonal real matrix A both these condition numbers are minimized when D is chosen such that the magnitudes of corresponding suband super-diagonal elements are equal. | If a tri-diagonal matrix A is such that corresponding suband super-diagonal elements have equal magnitude then A is said to be balanced or equilibrated. Wilkinson [5, p. 424] uses norms of balanced tri-diagonal matrices for error analysis of the eigenvalue problem. He observes that, given a tri-diagonal matrix A = [an] all of whose suband super-diagonal elements are nonzero, a diagonal matrix D = diag (di, d! ¿=1 \y{ Xi\ is minimized when D\ = I. This implies the corollary. Theorem 3. If A is a balanced tri-diagonal real matrix with distinct eigenvalues then A has an eigenvector matrix X = [x\, x-i, • • -, xn] such that k2(X) = mfj>1>D,fc2(Z>f1XDi). Proof. Bauer [2] showed that inf fc2(Z»r1XZ)2) ^ p(E1X-1EtX) Dl,D2 for all diagonal matrices E\ and E2 for which |2?i| = \E2\ = I (p denotes the spectral radius). Hence it suffices for us to obtain equality for some eigenvector matrix X of A and for some such E\ and E%. Let Q be a unitary matrix such that if Z = XQ then J = Z~lAZ is the direct sum of 1 by 1 and 2 by 2 matrices. (The latter are of the form License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use EXTREMAL PROPERTIES OF BALANCED TRI-DIAGONAL MATRICES 195 [-Î Û and correspond to conjugate complex pairs of eigenvalues X ± ip.) If the permutation matrix P is chosen such that X = XP, invariance of k2 implies that for all Di and D2 k(DrlXD2) = k(DrlXD2) = kiDr'XDi) = kiD^XPD,) = fc(Dl-1X(Pi)î¡P2,)) . Hence no generality is lost if we assume that those pairs of diagonal elements of D2 are equal which correspond to a complex conjugate pair of eigenvectors. Under this assumption k(D1~iXD2) = k(DrlXD2Q) = k(Dr'XQD2) , which allows us to replace the problem of minimizing k(Di~1XD2) by that of finding mîDl,D2 kiD^ZDi). Now Z^AZ = / implies ZtAtZ-t = jT = ElJEl for some real diagonal matrix E\ such that \E\\ = I. Hence, if AT = E%AE2, it follows that E2Z~TE\ — ZD2 for some diagonal matrix D2. Thus there exists a matrix Z0 such that Z0-IAZ0 = J as well as Z0_1 = EiZoTE2. Hence k(Z0) = llZolMI-EiZo^íllí = ll^olh2 = p(ZotZ0) = p(E1Z0-iE2Z0) . The result of Bauer stated at the beginning of this proof now establishes the theorem. Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey 07974 1. F. L. Bauer, "Some aspects of scaling invariance," Colloq. Internat. C.N.R.S., No. 165, pp. 37-47. 2. F. L. Bauer, "Optimally scaled matrices," Numer. Math., v. 5, 1963, pp. 73-87. MR 28 #2629. 3. E. E. Osborne, "On pre-conditioning of matrices," J. Assoc. Corrvput. Mach., v. 7, 1960, pp. 338-345. MR 26 #892. 4. J. Stoer & Ch. Witzgall, "Transformations by diagonal matrices in a normed space," Numer. Math., v. 4, 1962, pp. 158-171. MR 27 #154. 5. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use