Reduction of a dynamics and separated angles

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 A_m 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.

KEYWORDS:

• Reducible sequences
• separated angles
• well conditioning

2010 Mathematics Subject Classification:

• Primary: 37D99

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.

Additional information

Funding

This work was partially supported by FCT/Portugal through UID/MAT/04459/2013.