Also, if and are positive definite real matrices, then we get (2.5).
We reformulate (10) as the form of matrices, then we obtain r n ̂ ( f j ) = Φ j b n, j n = 1, 2, …, N (12).
Furthermore, if A and B are commutative ((AB=BA), A and B are (m times m) matrices), then we have expbigl{ (A+B W t bigr} =expbigl{ BW t bigr} expbigl{ AW t bigr}.
If and are -vectors of real numbers such that for some doubly stochastic matrix, then we say that is (vector) majorized by ; see [1].
If we consider some special cases of the matrix, then we have the following: (1) If A = C 1, the Cesaro matrix, then the definition reduces to ( B Λ μ ) n -statistical convergence.
If there exists a Jacket matrix J such that A = J ∑ J−1, where Σ is a diagonal matrix, then we say that A is a Jacket matrix similar to the diagonal matrix ∑.
If we replace the matrix A in Theorem 2.1 by identity matrix, then we immediately get the following result which is due to Erkuş and Duman [8]: Corollay 2.2.
Let (A=operatorname{Circ}(F_{0}, F_{1}, ldots, F_{n-1})) be a circulant matrix, then we have |A|_{2}leq F_{n}F_{n-1}, where (|cdot|_{2}) is the spectral norm and (F_{n}) denotes the nth Fibonacci number.
Let (A=operatorname{SCirc}(F_{0}, F_{1}, ldots, F_{n-1})) be a symmetric circulant matrix, then we have |A|_{2}leqsqrt{ n-1)F_{n}F_{n-1}}, where (|cdot|_{2}) is the spectral norm and (F_{n}) denotes the nth Fibonacci number.
Since A = ( a m k ) is a non-negative regular matrix, then we have 1 = lim m → ∞ ∑ k = 1 ∞ a m k = lim m → ∞ ∑ k ∉ K a m k + lim m → ∞ ∑ k ∈ K a m k.
Let (A=operatorname{Circ}(F_{0}, F_{1}, ldots, F_{n-1})) be a circulant matrix, then we have |A|_{2}leqsqrt{ n-1)F_{n}F_{n-1}}, where (|cdot|_{2}) is the spectral norm and (F_{n}) denotes the nth Fibonacci number.
