A. Cantoni and F. Bulter (1976), `Eigenvalues and eigenvectors of symmetric centrosymmetric matrices', Lin. Alg. Appl. 13, 275-288.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....case, the correlation matrices R yy [k] and R vv [k] are symmetric Toeplitz matrices. These matrices belong to the class of double symmetric matrices, which are symmetric with respect to both the main and the secondary diagonal and whose eigenvectors have special symmetry properties [29], i.e. every eigenvector is either symmetric or skew symmetric. Theorem 1: If W is constructed according to (27) then W satis es W = J W J (W J) 44) with J = J the M M reverse identity matrix. These properties hold in the white noise case as well as in the coloured noise case for ....

....is an eigenvalue decomposition. Because R yy [k] and R vv [k] are double symmetric matrices, J R yy [k] J = R yy [k] J R vv [k] J = R vv [k] 46) such that also yy [k] R vv [k] J R yy [k] R vv [k] J : 47) Therefore the eigenvectors, which are the columns of , satisfy the property [29] J diagf 1g ; 48) such that J W J = J J (49) W : 50) These symmetry properties imply that the ith row column of W is equal to the (L 1 i)th row column in reversed order. For L odd, the middle column in W is symmetric, and hence represents a linear phase ....

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P. Butler and A. Cantoni, \Eigenvalues and eigenvectors of symmetric centrosymmetric matrices," Linear Algebra and its Applications, vol. 13, pp. 275-288, Mar. 1976.

....the determinant, inverse, and eigenvalue and eigenvectors. Therefore, the MKM matrix has symmetric centrosymmetric structure if the domain possesses the symmetric geometry such as rectangle and ellipse. Such matrix structure makes the MKM conserve many important physical properties of the system [24]. The difference between the present methodology and the Fasshauer s Hermite method lies in that we collocates the governing and boundary equations separately at each boundary node. We name the present RBF collocation scheme as the modified Kansa s method to differentiate it from the other ....

.... local symmetric geometry partitioning, the factorization merit of symmetric centrosymmetric matrix can lead to further significantly reducing the computational effort, preserving the physical features of real system and improving computation stability in the FKM computing large size problems [24]. 3.3. Direct MKM and FKM schemes According to interpolation expression (41) the approximation solutions to u at all nodes of domain and boundary and to the Neumann boundary conditions at all boundary nodes can be expressed as = u , 54) L n n L n j j 2 2 , 55) n u = ....

Cantoni, A. and Butler, P., Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl. 13, 1976, 275-288.

....(ffi i;n 1 Gammai ) i;j=1; n denote the (n; n) flipmatrix with ones in its secondary diagonal and zeros elsewhere. A vector x 2 IR n is called symmetric if x = J n x, and it is called skew symmetric (or anti symmetric) if x = GammaJ n x. It is well known (cf. Andrew [1] Cantoni and Butler [2]) even for the larger class of symmetric and centrosymmetric matrices that every eigenspace of a matrix C n in this class (i.e. c n 1 Gammai;n 1 Gammaj = c ij for every i; j = 1; n) has a basis of vectors which are either symmetric or skew symmetric, and that there exists a basis of IR ....

....respectively. In [18] and [19] we improved the methods of [11] and [12] for computing the smallest eigenvalue of a symmetric Toeplitz matrix T n using even and odd secular equations and hence exploiting the symmetry properties of the eigenvectors of T n . Andrew [1] Cantoni and Butler [2] and Trench [16] the latter for symmetric Toeplitz matrices only) took advantage of symmetry properties to reduce the eigenvalue problem for C n to two eigenproblems of half the dimension. For a symmetric Toeplitz matrix T n 2 IR n , T n = t ji Gammajj ) their approach is as follows: If n = ....

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A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Lin. Alg. Appl. 13 (1976), pp. 275 --- 288

....in (13) can be arbitrary but is fixed. This kind of explicit formulation is robust, but it induces considerable redundancy. For example, it is well known that symmetric centro symmetric matrices have special properties, i.e. dn=2e of the eigenvectors are symmetric and bn=2c are skew symmetric [7]. In the above, a vector v 2 R n is said to be symmetric (or skew symmetric) if Ev = v (or Ev = Gammav) where E = e ij ) 2 R n Thetan is the exchange matrix defined by e ij = 1; if i j = n 1; 0; otherwise. Symmetric Toeplitz matrices are symmetric and centro symmetric. However, the ....

A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl., 13(1976), 275-288.

.... Phi s k Delta s T k Psi are symmetric Toeplitz matrices. Symmetric Toeplitz matrices belong to the class of double symmetric matrices, which are symmetric about both the main diagonal and the secondary diagonal. The eigenvectors of such matrices are known to have special symmetry properties [7]. Using these properties, one can prove the following symmetry property for the filter WWF . Theorem 1 If WWF is constructed according to equation (9) then WWF satisfies the symmetry properties WWF = J Delta WWF Delta J (12) W T WF = J Delta W T WF Delta J; 13) with J the reverse ....

P. Butler and A. Cantoni, "Eigenvalues and eigenvectors of symmetric centrosymmetric matrices," Linear Algebra and its Applications, vol. 13, pp. 275--288, Mar. 1976.

.... Phi s k Delta s T k Psi are symmetric Toeplitz matrices. Symmetric Toeplitz matrices belong to the class of double symmetric matrices, which are symmetric about both the main diagonal and the secondary diagonal. The eigenvectors of such matrices are known to have special symmetry properties [6]. Using these properties, one can prove the following symmetry property for the optimal filter WWF [7] Theorem 1 If WWF is constructed according to equation (9) then WWF satisfies the symmetry properties WWF = J Delta WWF Delta J (12) W T WF = J Delta W T WF Delta J; 13) with J ....

P. Butler and A. Cantoni, "Eigenvalues and eigenvectors of symmetric centrosymmetric matrices", Linear Algebra and its Applications, vol. 13, pp. 275--288, Mar. 1976.

.... (N Gamma 2) N Gamma 3) 0) 3 7 7 7 7 7 5 : 2.40) Symmetric Toeplitz matrices belong to the class of double symmetric matrices, which are symmetric about both the main diagonal and the secondary diagonal. The eigenvectors of such matrices are known to have special symmetry properties [17][18] For specific notation and properties, we refer to appendix B. Theorem 1 If the filter WWF is constructed according to equations (2.19) 2.24) then WWF satisfies WWF = J Delta WWF Delta J (2.41) W T WF = J Delta W T WF Delta J (2.42) with J a matrix with all ones along its ....

.... an eigenvalue 1 with multiplicity N ( Phi uu and Phi nn have N eigenvalues which are the same) The eigenspace corresponding to this eigenvalue consists of N eigenvectors which are a linear combination of symmetric and skew symmetric vectors, and hence, are neither symmetric nor skew symmetric [17]. Therefore the general class of estimators exhibits no symmetry properties at all. However, if we only retain the N eigenvectors X 1 which are symmetric or skewsymmetric and discard the N eigenvectors X 2 which are neither symmetric nor skewsymmetric, then we can prove the same symmetry ....

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P. Butler and A. Cantoni, "Eigenvalues and eigenvectors of symmetric centrosymmetric matrices," Linear Algebra and its Applications, vol. 13, pp. 275-- 288, Mar. 1976.

....is assumed to be clear from the context. 2 Preliminaries A symmetric matrix T 2 IR (n;n) is said to be Toeplitz if its elements T ij satisfy T ij = t jj Gammaij for some vector t = t 0 ; t n Gamma1 ) T 2 IR n . Many early results about such matrices can be found in, e.g. 3] [6] and [9] Toeplitz matrices are persymmetric, i.e. they are symmetric about their southwestnortheast diagonal. For such a matrix T , this is the same as requiring that JT T J = T , where J is a matrix with ones on its southwest northeast diagonal and zeros everywhere else (the n Theta n ....

....and persymmetric is called doubly symmetric. A symmetric vector v is defined as a vector satisfying Jv = v and an antisymmetric vector w as one that satisfies Jw = Gammaw. If these vectors are eigenvectors, then their associated eigenvalues are called even and odd, respectively. It was shown in [6] that, given a real symmetric Toeplitz matrix T of order n, there exists an orthonormal basis for IR n , composed of n Gamma bn=2c symmetric and bn=2c antisymmetric eigenvectors of T , where bffc denotes the integral part of ff. In the case of simple eigenvalues, this is easy to see from the ....

Cantoni, A. and Butler, F. (1976): Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl., 13, pp. 275-288.

.... x 0 6= 0 T Gamma1 n = 1 x 0 i Ln (x)Un (y) Gamma Ln (ZJy)Un (ZJx) j (4) we get V = span(x; ZJy) and VT = span(y; ZJx) with J the counteridentity defined by J i;j = ffi n 1 Gammaj;j for i; j = 0; n and T GammaT n = JT Gamma1 n J , because Tn and T Gamma1 n are persymmetric [3]. The vectors x and ZJy are not linear independent iff T Gamma1 n e 0 = ZJT GammaT n e 0 for a real number , or 0 = Z T TnZ) T Gamma1 n Je 0 ) Tn Gamma e n (t n : t 0 ) Gamma (t Gamman : t Gamma1 0) T e T n )T Gamma1 n e n = e n Gamma e n e T n e n Gamma Z T ....

Cantoni,A.,Butler,P.: Eigenvalues and Eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl. 13, 275-288, 1976.

....In Section 6, we present numerical results. 2 Preliminaries A symmetric matrix T 2 IR (n;n) is said to be Toeplitz if its elements T ij satisfy T ij = t jj Gammaij for some vector t = t 0 ; t n Gamma1 ) T 2 IR n . Many early results about such matrices can be found in, e.g. 3] [6] and [10] Toeplitz matrices are persymmetric, i.e. they are symmetric about their southwestnortheast diagonal. For such a matrix T , this is the same as requiring that JT T J = T , where J is a matrix with ones on its southwest northeast diagonal and zeros everywhere else (the exchange ....

....symmetric vector v is defined as a vector satisfying Jv = v and a skew symmetric or antisymmetric vector w as one that satisfies Jw = Gammaw. If these vectors are eigenvectors, then their associated eigenvalues are called even and odd, respectively. Drawing on the results in [3] it was shown in [6] that, given a real symmetric Toeplitz matrix T of order n, there exists an orthonormal basis for IR n , composed of n Gamma bn=2c symmetric and bn=2c skew symmetric eigenvectors of T , where bffc denotes the integral part of ff. In the case of simple eigenvalues, this is easy to see from the ....

Cantoni, A. and Butler, F. (1976): Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl., 13, pp. 275-288.

....signed permutation matrix) Chen88] ChSa89a] On the other hand, a matrix reflexive with respect to some reflection matrix P is always a circulative matrix since any reflection matrix is necessarily a circulation matrix. The class of centrosymmetric matrices [Aitk49] Andr73a] Andr73b] [Cant76] is of course a special class of circulative matrices because it is a special class of reflexive matrices. In this section, we give another important special case. Let U 1 (x) 1 and U 2 (x) and U 3 (x) x = 2 =k for some positive integer k, be the following two orthogonal matrices obtained from ....

A. Cantoni, Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices, Linear Algebra and Its Applications 13, 275-288 (1976).

....determined by a vector t = t 0 ; t n Gamma1 ) T 2 IR n . In what follows, we consider only symmetric matrices, which exhibit some special properties that will be useful later on and which we now briefly present. Many early results about such matrices can be found in, e.g. 4] [8] and [14] Toeplitz matrices belong to the larger class of persymmetric matrices, i.e. matrices symmetric about their southwest northeast diagonal. For such a matrix T , this is the same as requiring that JT T J = T , where J is a matrix with ones on its southwestnortheast diagonal and zeros ....

....of a persymmetric matrix is also persymmetric. We define a symmetric vector v as a vector satisfying Jv = v and a skew symmetric vector w as one that satisfies Jw = Gammaw. If these vectors are eigenvectors, then the associated eigenvalues are called even and odd, respectively. It was shown in [8] that, given a real symmetric Toeplitz matrix T of order n, there exists an orthonormal basis for IR n , composed of n Gamma bn=2c symmetric and bn=2c skew symmetric eigenvectors of T , where bffc denotes the integral part of ff. In the case of simple eigenvalues, this is easy to see from the ....

Cantoni, A. and Butler, F. (1976): Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl., 13, pp. 275-288.

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A. Cantoni and F. Bulter (1976), `Eigenvalues and eigenvectors of symmetric centrosymmetric matrices', Lin. Alg. Appl. 13, 275-288.

No context found.

A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl., 13(1976), 275-288.

No context found.

A. Cantoni and F. Bulter. Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl., 13:275--288, 1976.

No context found.

A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl., 13(1976), 275-288.

No context found.

A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl., 13(1976), 275-288.

No context found.

A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Alg. Appl., 13(1976), 275-288.

No context found.

A. Cantoni and P. Butler, "Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices," Linear Algebra and Its Applications, vol. 13, pp. 275--288, 1976.

No context found.

A. Cantoni and P. Butler, "Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices," Linear Algebra and Its Applications, vol. 13, pp. 275--288, 1976.

No context found.

A. Cantoni and P. Butler, "Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices," Linear Algebra and Its Applications, vol. 13, pp. 275--288, 1976.

No context found.

A. Cantoni and P. Butler, "Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices," Linear Algebra and Its Applications, vol. 13, pp. 275--288, 1976.

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