The Largest Eigenvalue of Nonnegative Tensors
The Largest Eigenvalue of Nonnegative Tensors
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发表时间：
2013-09
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通讯作者：
Nur Fadhilah Ibrahim
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作者：
Nur Fadhilah Ibrahim
张量只是矩阵的概括。矩阵的许多属性已推广到张量。在过去的几年里，张量谱理论得到了发展。 Perron-Frobenius 定理和极小极大定理是非负矩阵性质已扩展到非负张量的两个例子。这导致将 Collatz 方法扩展到非负张量，以查找非负矩阵的最大特征值。本论文研究了求正方形张量和矩形张量最大特征值的方法。我们还研究了方法的收敛性，证明了矩形张量方法在弱不可约性条件下是Q线性收敛的。我们进一步将该方法推广到非负多项式特征值问题。该方法对于不可约非负多项式是收敛的。我们探讨齐次和非齐次多项式的情况。我们还提出了一种收敛方法来解决优化问题，其中目标函数是具有球形约束的非负一般多项式。
Tensors are simply generalisation of matrices. Many properties of matrices have been generalised to tensors. Over the past few years, the spectral theory of tensors has been developed. The Perron-Frobenius Theorem and the minimax theorem are two examples of the property of nonnegative matrices which have been extended to nonnegative tensors. This leads to extension of the Collatz method for finding the largest eigenvalue of nonnegative matrices to nonnegative tensors. In this thesis, we study the methods for finding the largest eigenvalue of square tensors and rectangular tensors. We also study the convergence of the methods and show that the method for rectangular tensors is Q-linear convergence under weak irreducibility condition. We further generalise the method to nonnegative polynomial eigenvalue problems. The method is convergent for irreducible nonnegative polynomials. We explore the case for both homogeneous and nonhomogeneous polynomials. We also present a convergent method for solving the optimisation problem where the objective function is a nonnegative general polynomial with spherical constraint.