ABSTRACT
We show that the notions of well conditioning and of separated angles are equivalent for any non-autonomous dynamics with discrete time defined by a sequence of matrices. Equivalently, for a sequence of matrices Am the ratio between any axes of the ellipsoid , where
and B is the unit ball centred at the origin, is bounded in m if and only if the angles
are bounded from below and above in m for any non-collinear non-zero vectors v1 and v2. As a non-trivial consequence, we show that any sequence of matrices with separated angles can be reduced by a bounded non-autonomous coordinate change to one whose matrices are multiples of the identity. Moreover, we consider the problem of whether two given sequences of matrices can be reduced one to another, both when they have separated angles and when they have not.
2010 Mathematics Subject Classification:
Acknowledgements
This work was partially supported by FCT/Portugal through UID/MAT/04459/2013.
Disclosure statement
No potential conflict of interest was reported by the authors.