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On the \(\lambda \)-robustness of matrices over fuzzy algebra. (English) Zbl 1225.15027
Let \((B,\leq)\) be a non-empty, bounded, linearly ordered set. Define the operations \(a\oplus b=\max\{a,b\}\) and \(a\otimes b=\min\{a,b\}\) for \(a,b\in B\). Let \(A=[a_{ij}]_{n\times n}\) be a square matrix with coefficients in \(B\). A column vector \(x\in B^n\) is said to be a \(\lambda\)-eigenvector of \(A\) for some \(\lambda\in B\) if \(A\otimes x=\lambda\otimes x\).
The matrix \(A\) is called \(\lambda\)-robust if for every \(x\in B^n\) the vector \(A^k\otimes x\) is a \(\lambda\)-eigenvector of \(A\) for some \(k\in{\mathbb Z}^+\). Let \(V(A,\lambda)\) denote the set of all \(\lambda\)-eigenvectors of \(A\). The authors show that: When \(\lambda\geq\max\{ a_{ij}: 1\leq i,j\leq n\}\), \(A\) is \(\lambda\)-robust if and only if \(V(A,\lambda)=V(A^{\ell},\lambda)\) for each \(\ell\in{\mathbb Z}^+\). Note that \(V(A^{\ell},\lambda)=V(A^{\ell},I)\) for \(\ell\in{\mathbb Z}^+\) whenever \(\lambda\geq\max\{ a_{ij}: 1\leq i,j\leq n\}\). An \(O(n^3)\) time algorithm exists to decide whether \(A\) is \(\lambda\)-robust.
Let \(M(A)\) denote the set of all vectors \(x=[x_i]_{n\times 1}\in B^n\) with each \(x_i<c(A)\) for \(c(A)=\bigotimes_{i=1}^n\left(\bigoplus_{j=1}^n a_{ij}\right)\). The matrix \(A\) is called strongly \(\lambda\)-robust if for every \(x\in B^n\backslash M(A)\) the vector \(A^k\otimes x\) is the greatest \(\lambda\)-eigenvector \(\bigoplus_{y\in V(A,\lambda)}y\) of \(A\) for some \(k\in{\mathbb Z}^+\). A main result of the paper gives equivalent conditions for \(A\) being strongly \(\lambda\)-robust when \(\lambda > c(A)\). Details are too involved to describe here. Basing on this, an \(O(n^3)\) algorithm is introduced to decide whether \(A\) is strongly \(\lambda\)-robust.

15B15 Fuzzy matrices
15A18 Eigenvalues, singular values, and eigenvectors
65F30 Other matrix algorithms (MSC2010)
Full Text: DOI
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