Abstract
In this paper, based on combinatorial methods and the structure of RFMLR-circulant matrices, we study the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions. Firstly, we give some properties of exponential forms and trigonometric functions. Furthermore, we study Frobenius norms, the lower and upper bounds for the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions by some ingenious algebra methods, and then we obtain new refined results.
Highlights
Matrix analysis theory is a powerful tool to study modern communication systems; especially, matrix norm is very important for neural network-based adaptive tracking control for switched nonlinear systems [1]
Conclusion e spectral norms of row first-minus-last right- (RFMLR-)circulant matrices involving exponential forms and trigonometric functions are investigated in this paper. e computation complexity of this paper is lower than the previous work
By using the algorithms of this paper, we can further study the identities of f(x)-circulant matrices, such as RFPrLrR-cieculant matrix and RFMLrR-circulant matrix
Summary
Matrix analysis theory is a powerful tool to study modern communication systems; especially, matrix norm is very important for neural network-based adaptive tracking control for switched nonlinear systems [1]. For e(x), e(x) e2πix and |e(x)| 1; by eiθ cos θ + i sin θ, note that e(0) e(1) e(− 1) e(n) 1, and by the trigonometric sums, we have n km ⎧⎨ n, n|m, By the relationship between exponential forms e(k/n) and trigonometric functions cos(kπ/n) and sin(kπ/n), we can obtain some power sums of these functions. A1 a2 − a1 a3 − a2 · · · an− 1 − an− 2 a0 − an− 1 n×n (2)
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