
Isospectral Matrix Flows for Numerical Analysis
43
.,4 = [A, (log A)_].
That is, for any k E N0,
A(k)
= k-
th step in the qR-algorithm applied to A(0).
Similar results also exist for versions of the QR-algorithm incorporating shifts; see Watkins
and Elsner [32].
S THE DOUBLE BRACKET FLOW
Brockett [5] has introduced a new class of isospectral flows on the set of symmetric matrices.
He considers the double Lie bracket equation
)(
= [X,[X,A]]
= AX 2 + X2A- 2XAX
(27)
where
[A, B] = AB-BA
denote the Lie bracket and A, X are real symmetric n x n matrices.
Here the matrix A plays the role of a parameter to guide ...