Spectral radius

Summary

In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·). Definition Matrices Let λ1, ..., λn be the eigenvalues of a matrix A ∈ Cn×n. The spectral radius of A is defined as :\rho(A) = \max \left { |\lambda_1|, \dotsc, |\lambda_n| \right }. The spectral radius can be thought of as an infimum of all norms of a matrix. Indeed, on the one hand, \rho(A) \leqslant |A| for every natural matrix norm |\cdot|; and on the other hand, Gelfand's formula states that \rho(A) = \lim_{k\to\infty} |A^k|^{1/k} . Both of these results are shown below. However, the spectral radius does not necessarily satisfy |A\mathbf{v}| \leqslant \rho(A) |\mathbf{v}| for arbitrary

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