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I am trying to figure out how to calculate this quantity:

$$ \frac{\lVert U^{t}_{\mathbf{x_0}}\mathbf{e}_1\wedge U^{t}_{\mathbf{x_0}}\mathbf{e}_2\wedge\ldots\wedge U^{t}_{\mathbf{x_0}}\mathbf{e}_k\rVert}{\lVert \mathbf{e}_1\wedge \mathbf{e}_2\wedge\ldots\wedge \mathbf{e}_k\rVert} $$

where $\wedge$ is an exterior product and $\lVert\circ\rVert$ is a norm with respect to some Riemannian metric. I can calculate the $U^{t}_{\mathbf{x_0}}\mathbf{e}_i$ vectors but can't figure out how to proceed after that.
How would one define this norm of an exterior product and how to calculate it numerically? This is from,

Ippei Shimada and Tomomasa Nagashima
A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems Prog. Theor. Phys. (1979) 61 (6): 1605-1616 doi:10.1143/PTP.61.1605

an algorithm to calculate Lyapunov spectra.

m3phi
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  • Hint: write these wedge products in terms of a set of orthogonal (or orthonormal) $k$-vectors. If the dimension of the space is $k$, this should simplify considerably. Can you write out a basis of orthogonal (or orthonormal) $k$-vectors? – Muphrid Feb 19 '15 at 06:51

1 Answers1

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In k-dimensional space this is simply $|\det U|$; for k-parallelotope $\sqrt{\det(U^TU)}$, see a related "Ratio of area formed by transformed and original sides of a parallelogram".

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rych
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