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Spectral properties of left triangular matrices. (English. Russian original) Zbl 0997.15008
Russ. Math. Surv. 56, No. 1, 149-151 (2001); translation from Usp. Mat. Nauk 56, No. 1, 155-156 (2001).
A left triangular \(n\times n\) matrix with binary \((0,1)\) entries has entries \(a_{jk}=1\) for \(1\leq j\leq n-k+1\) and 0 otherwise. For the family of these matrices with \(n\to\infty\) an asymptotically exact representation of a complete system of eigenvalues \(\lambda_k^{(n)}\) and eigenvectors \(e_k^{(n)}\) is obtained, namely, \[ \lambda_k^{(n)}= (-1)^{k+1} {n\over(k-\tfrac 12)\pi}+ 0\Bigl(\tfrac 1n\Bigr),\;(e_k^{(n)})_i= \cos\bigl[(k-\tfrac 12)(i-1) \pi/n \bigr]+0\bigl( \tfrac 1n\bigr). \] A similar result for a more general family of left triangular matrices with certain function values of a \(C^1\)-function \(\varphi:[0,1] \mapsto(0,1]\) as entries is also obtained. These results are applied to the analysis of ergodic properties of chaotic maps resulting from a piecewise linear function.
MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A30 Ergodic theorems, spectral theory, Markov operators
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