Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

The Condition Number of Similarities that Diagonalize Matrices

James W. Demmel

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-83-127
July 1983

http://www.eecs.berkeley.edu/Pubs/TechRpts/1983/CSD-83-127.pdf

How ill-conditioned must a matrix S be if it (block) diagonalizes a given matrix T, i.e. if S(-1) TS is block diagonal? The answer depends on how the diagonal blocks partition T's spectrum; the condition number of S is bounded below by a function of the norms of the projection matrices determined by the partitioning. In the case of two diagonal blocks we compute an S which attains this lower bound, and we describe almost best conditioned S's for dividing T into more blocks. We apply this result to bound the error in an algorithm to compute analytic functions of matrices, for instance exp( T). Our techniques also produce bounds for submatrices that appear in the square-root-free Choleskt and in the Gram-Schmidt orthogonalization algorithms.


BibTeX citation:

@techreport{Demmel:CSD-83-127,
    Author = {Demmel, James W.},
    Title = {The Condition Number of Similarities that Diagonalize Matrices},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1983},
    Month = {Jul},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1983/6332.html},
    Number = {UCB/CSD-83-127},
    Abstract = {How ill-conditioned must a matrix <i>S</i> be if it (block) diagonalizes a given matrix <i>T</i>, i.e. if <i>S</i>(-1)<i>TS</i> is block diagonal? The answer depends on how the diagonal blocks partition <i>T</i>'s spectrum; the condition number of <i>S</i> is bounded below by a function of the norms of the projection matrices determined by the partitioning. In the case of two diagonal blocks we compute an <i>S</i> which attains this lower bound, and we describe almost best conditioned <i>S</i>'s for dividing <i>T</i> into more blocks. We apply this result to bound the error in an algorithm to compute analytic functions of matrices, for instance exp(<i>T</i>). Our techniques also produce bounds for submatrices that appear in the square-root-free Choleskt and in the Gram-Schmidt orthogonalization algorithms.}
}

EndNote citation:

%0 Report
%A Demmel, James W.
%T The Condition Number of Similarities that Diagonalize Matrices
%I EECS Department, University of California, Berkeley
%D 1983
%@ UCB/CSD-83-127
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1983/6332.html
%F Demmel:CSD-83-127