Extremal Properties of Balanced Tri-Diagonal Matrices

Peter A. Businger
1969 Mathematics of Computation  
If A is a square matrix with distinct eigenvalues and D a nonsingular matrix, then the angles between row-and 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
more » ... or a tri-diagonal real matrix A both these condition numbers are minimized when D is chosen such that the magnitudes of corresponding sub-and super-diagonal elements are equal. | If a tri-diagonal matrix A is such that corresponding sub-and 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 sub-and super-diagonal elements are nonzero, a diagonal matrix D = diag (di, d<i, • • •, dn) can be found such that D~lAD is balanced. In fact, such a D is defined by di+i/di = (|ai+i,i|/|a¿,i+1|)1/2, i = 1,2, ■ ■ -,n -1 . If some sub-or super-diagonal element of A is zero then finding its eigenvalues can be reduced to finding the eigenvalues of submatrices, each of which can be balanced separately. It is an immediate consequence of Osborne's Lemma 2 [3] that a balanced tridiagonal matrix A has the extremal property \\A\\E = irá {¡D^ADWe , D where || • ||e denotes the Euclidean matrix norm (Schur norm, Frobenius norm). Our Theorem 1 states the analogous result for the spectral norm; Theorems 2 and 3 show that the eigenvalue problem of a balanced tri-diagonal matrix is optimally conditioned in the sense that no matrix of the form D~lAD has smaller angles between corresponding row-and column-eigenvectors. We use || • || to denote the Euclidean vector norm, ][ -J|2 for the subordinate matrix bound (the spectral matrix norm), kt(-) for the spectral condition number of a nonsingular matrix, and kE(-) for the Euclidean condition number (defined by kE(X) = \\X\\e ||X_1||b). Absolute value signs applied to vectors are understood componentwise. D, Di, and Di denote diagonal matrices with positive diagonal elements. Theorem 1. If A is a balanced tri-diagonal real matrix then \\A\U = mîWD-iADh . D
doi:10.2307/2005072 fatcat:idepmmhwn5bbhofftasus2jldi