# 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