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Additional info for A Stereo Vision System for Position Measurments and Recognition in an Autonomous Robotic System for Carrying Food Trays

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Nevertheless, the ideas behind the proofs may give some insight on what is actually difficult in computing a joint spectral radius, and where is the limit between feasibility and infeasibility. For these reasons we limit ourselves to present the main ideas of the theorems, and we provide bibliographic references for the interested reader. We add that for now on and unless explicitly stated we restrict our attention to finite sets of matrices. 1 NP-Hardness The first theorem we present is on the NP-hardness of the joint spectral radius approximation, and is valid even for binary matrices.

A1 x0 | < ε , |At j . . A1 x0 | = 1, for any ε . Setting z = At j . . 1 that A has an eigenvalue λ such that |1 − λ | = O(ε 1/n ), which implies that sup {ρ (A) : A ∈ Σ ∗ } ≥ 1. 5), one just has to apply the result to each irreducible block separately. This ends the first section of this chapter. We have now a fair insight on the behavior of the joint spectral radius: Given a set of matrices, there is always a set of submatrices that is irreducible, and whose joint spectral radius has the same value.

If the Lie Algebra associated to Σ is solvable, then Σ is commonly upper triangularizable. 5. Let Σ be a set of matrices. If Σ generates a solvable Lie algebra, then ρ (Σ ) = ρ1 (Σ ). 6): ρt (Σ ) ≤ ρ (Σ ) ≤ ρˆt (Σ ), remembering that the successive bounds given by the left hand side as well as the right hand side tend to ρ when t → ∞. So an immediate algorithm would be to fix a t, compute all products of length t, take the maximal spectral radius as a lower bound, and the maximal norm (for a fixed norm) as an upper bound.

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