Matrix Pencils pp 2-16 | Cite as

# The condition number of equivalence transformations that block diagonalize matrix pencils

## Abstract

How ill-conditioned must a matrix *S* be if its columns are constrained to span certain subspaces? We answer this question in order to find nearly best conditioned matrices *S*_{R} and *S*_{L} that block diagonalize a given matrix pencil *T*=*A*+λ*B*, i.e. *S*_{L}^{−1}*TS*_{R}=Θ is bloc diagonal. We show that the best conditioned *S*_{R} has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of Θ. Thus, the more nearly the right subspaces overlap the more ill-conditioned *S*_{R} must be. The same is true of *S*_{L} and the left subspaces. For the standard eigenproblem (*T*=*A*−λ*I*), *S*_{L} = *S*_{R} and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number, namely the norm of the projection matrix associated with one of the subspaces. We apply this result to bound the error in an algorithm to compute analytic functions of matrices, for instance exp(*T*).

## Preview

Unable to display preview. Download preview PDF.

### References

- [0]H. BART, I. GOHBERG, M. A. KAASHOEK, P. VAN DOOREN, Factorizations of Transfer Functions, SIAM J. Control, vol 18, no 6, November 1980, pp 675–696MathSciNetCrossRefMATHGoogle Scholar
- [1]F. L. BAUER, A further generalization of the Kantorovic inequality,
*Numer. Math.*, 3, pp 117–119, 1961MathSciNetCrossRefMATHGoogle Scholar - [2]F. L. BAUER, Optimally scaled matrices,
*Numer. Math.*, 5, pp. 73–87, 1983MathSciNetCrossRefMATHGoogle Scholar - [3]F. L. BAUER, C. T. FIKE, Norms and Exclusion Theorems,
*Numer. Math.*, 2, pp 137–141, 1960MathSciNetCrossRefMATHGoogle Scholar - [4]F. L. BAUER, A. S. HOUSEHOLDER, Some inequalities involving the euclidean condition of a matrix,
*Numer. Math.*, 2, pp 308–311, 1960MathSciNetCrossRefMATHGoogle Scholar - [5]C. DAVIS, W. KAHAN, Some new bounds on perturbations of subspaces, Bull. A. M. S., 75, 1969, pp 863–8MathSciNetCrossRefMATHGoogle Scholar
- [6]J. DEMMEL, The Condition Number of Similarities that Diagonalize Matrices, Electronics Research Laboratory Memorandum, University of California, Berkeley, 1982Google Scholar
- [7]F. R. GANTMACHER,
*The Theory of Matrices*, trans. K. A. Hirsch, Chelsea, 1959, vol. 2Google Scholar - [8]E. ISAACSON, H. B. KELLER,
*Analysis of Numerical Methods*, Wiley, 1966Google Scholar - [9]W. KAHAN, Conserving Confluence Curbs Ill-Condition, Technical Report 6, Computer Science Dept., University of California, Berkeley, August 4, 1972Google Scholar
- [10]T. KATO, Estimation of Iterated Matrices, with Application to the von Neumann Condition,
*Numer. Math.*, 2, pp 22–29, 1980MathSciNetCrossRefMATHGoogle Scholar - [11]T. KATO,
*Perturbation Theory for Linear Operators*, Springer-Verlag, 1966Google Scholar - [12]A. RUHE, Properties of a Matrix with a Very Ill-conditioned Eigenproblem,
*Numer. Math.*, 15, pp 57–60, 1970MathSciNetCrossRefMATHGoogle Scholar - [13]A. VAN DER SLUIS, Condition Numbers and Equilibration of Matrices,
*Numer. Math.*, 14, pp 14–23, 1969MathSciNetCrossRefMATHGoogle Scholar - [14]R. A. SMITH, The Condition Numbers of the Matrix Eigenvalue Problem,
*Numer. Math.*, 10, pp 232–240, 1967MathSciNetCrossRefMATHGoogle Scholar - [15]G. W. STEWART, Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems,
*SIAM Review*, vol. 15, no. 4, Oct 1973, p 752MathSciNetCrossRefGoogle Scholar - [16]J. G. SUN, The Perturbation Bounds for Eigenspaces of Definite Matrix Pairs, to appearGoogle Scholar
- [17]P. VAN DOOREN, P. DEWILDE, Minimal Cascade Factorization of Real and Complex Rational Transfer Matrices,
*IEEE Trans. on Circuits*and*Systems*, vol. CAS-28, no. 5, May 1981, p 395.MathSciNetMATHGoogle Scholar - [18]J. H. WILKINSON,
*Rounding Errors in Algebraic Processes*, Prentice Hall, 1963Google Scholar - [19]J. H. WILKINSON,
*The Algebraic Eigenvalue Problem*, Oxford University Press, 1965Google Scholar - [20]J. H. WILKINSON, Note on Matrices with a Very Ill-Conditioned Eigenproblem,
*Numer. Math.*, 19, pp 176–178, 1972MathSciNetCrossRefMATHGoogle Scholar