In this paper,a new matrix decomposition called the weighted polar decomposition is considered.Two uniqueness theorems of weighted polar decomposition are presented,and the best approximation property of weighted unit...In this paper,a new matrix decomposition called the weighted polar decomposition is considered.Two uniqueness theorems of weighted polar decomposition are presented,and the best approximation property of weighted unitary polar factor and perturbation bounds for weighted polar decomposition are also studied.展开更多
This study aims to investigate the polar decomposition of tensors with the Einstein product for thefirst time.The polar decomposition of tensors can be computed using the singular value decomposition of the tensors wit...This study aims to investigate the polar decomposition of tensors with the Einstein product for thefirst time.The polar decomposition of tensors can be computed using the singular value decomposition of the tensors with the Einstein product.In the following,some iterative methods forfinding the polar decomposi-tion of matrices have been developed into iterative methods to compute the polar decomposition of tensors.Then,we propose a novel parametric iterative method tofind the polar decomposition of tensors.Under the obtained conditions,we prove that the proposed parametric method has the order of convergence four.In every iteration of the proposed method,only four Einstein products are required,while other iterative methods need to calculate multiple Einstein products and one tensor inversion in each iteration.Thus,the new method is superior in terms of efficiency index.Finally,the numerical comparisons performed among several well-known methods,show that the proposed method is remarkably efficient and accurate.展开更多
The physical decomposition method separates atmospheric variables into four parts, correlating each with solar radiation, land-sea distribution, and inter-annual and seasonal internal forcing, strengthening the anomal...The physical decomposition method separates atmospheric variables into four parts, correlating each with solar radiation, land-sea distribution, and inter-annual and seasonal internal forcing, strengthening the anomaly signal and increasing the correlation between variables. This method was applied to the reanalysis data from the National Centers for Environmental Prediction/National Center for Atmospheric Research （NCEP/NCAR）, to study the effects of Arctic factors （Arctic oscillation （AO） and Arctic polar vortex） on wintertime temperatures in the Northern Hemisphere and China. It was fotmd that AO effects on zonal average temperature disturbance could persist for 1 month. In the AO negative phase in wintertime, the temperatures are lower in the mid-high latitudes than in normal years, but higher in low latitudes. When the polar vortex area is bigger, the zonal average temperature is lower at 50N. Influenced mainly by meridional circulation enhancement, cold air flows from high to low latitudes; thus, the temperatures in Continental Europe and the North American continent exhibit an antiphase seesaw relationship. When the AO is in negative phase and the Arctic polar vortex larger, the temperature is lower in Siberia, but higher in Greenland and the Bering Strait. Influenced by westerly troughs and ridges, the polar air disperses mainly along the tracks of atmospheric activity centers. The AO index can be considered a predictor of wintertime temperature in China. When the AO is in negative phase or the Asian polar vortex is intensified, temperatures in Northeast China and Inner Mongolia are lower, because under the influence of the Siberia High and northeast cold vortex, the cold air flows southwards.展开更多
In this paper, we prove that the norm closure of all linear combinations of two unitary operators is equal to the norm closure of all invertible operators in B(H). We apply the results to frame representations and giv...In this paper, we prove that the norm closure of all linear combinations of two unitary operators is equal to the norm closure of all invertible operators in B(H). We apply the results to frame representations and give some simple and alternative proofis of the propositions in 'P. G. Casazza, Every frame is a sum of three (but not two) orthonormal bases-and other frame representations, J. Fourier Anal. Appl., 4(6)(1998), 727-732.'展开更多
IN ref. [1] for the subspaces of Hilbert space H the concept of equivalence and a generalizeddimension are introduced. Note that in refs. [2, 3] many results are true only in the separablespace. We introduce a concept...IN ref. [1] for the subspaces of Hilbert space H the concept of equivalence and a generalizeddimension are introduced. Note that in refs. [2, 3] many results are true only in the separablespace. We introduce a concept of double unitary equivalence from the singular decomposition ofmatrix. In this note we get the conditions of doub1e unitary equivalent operators. Some gener-al results are obtained. It is well known that two operators A and B in Hilbert space H are called unitary equiva-lence if there is an unitary operator U such that A = U~* BU. In this case A and B can beconsidered as two complete same operators. Therefore, the condition of two-operator uitaryequivalence is very high. In the matrix theory there is a singular展开更多
Let A be an m×n complex matrix. A decomposition A=QH is termed a polar decomposition of A if Q *Q=I and H is a positive semidefinite Hermitian matrix. Perturbation bounds for Q when A and its perturbation are of ...Let A be an m×n complex matrix. A decomposition A=QH is termed a polar decomposition of A if Q *Q=I and H is a positive semidefinite Hermitian matrix. Perturbation bounds for Q when A and its perturbation are of non-full rank matrices are provided.展开更多
Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformatio...Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.展开更多
In this paper,we study the perturbation bounds for the polar decomposition A=QH where Q is unitary and H is Hermitian.The optimal (asymptotic) bounds obtained in previous works for the unitary factor,the Hermitian fac...In this paper,we study the perturbation bounds for the polar decomposition A=QH where Q is unitary and H is Hermitian.The optimal (asymptotic) bounds obtained in previous works for the unitary factor,the Hermitian factor and singular values of A areσ_r^2||ΔQ||_F^2≤||ΔA||_F^2, 1/2||ΔH||_F^2≤||ΔA||_F^2 and ||Δ∑||_F^2≤||ΔA||_F^2,respectively,where∑=diag(σ_1,σ_2,...,σ_r,0,...,0) is the singular value matrix of A andσ_r denotes the smallest nonzero singular value.Here we present some new combined (asymptotic) perturbation boundsσ_r^2||ΔQ||_F^2+1/2||ΔH||_F^2≤||ΔA||_F^2 andσ_r^2||ΔQ||_F^2+||Δ∑||_F^2≤||ΔA||_F^2 which are optimal for each factor.Some corresponding absolute perturbation bounds are also given.展开更多
文摘In this paper,a new matrix decomposition called the weighted polar decomposition is considered.Two uniqueness theorems of weighted polar decomposition are presented,and the best approximation property of weighted unitary polar factor and perturbation bounds for weighted polar decomposition are also studied.
基金funded by Iran National Science Foundation(INSF)under project No.4013447.
文摘This study aims to investigate the polar decomposition of tensors with the Einstein product for thefirst time.The polar decomposition of tensors can be computed using the singular value decomposition of the tensors with the Einstein product.In the following,some iterative methods forfinding the polar decomposi-tion of matrices have been developed into iterative methods to compute the polar decomposition of tensors.Then,we propose a novel parametric iterative method tofind the polar decomposition of tensors.Under the obtained conditions,we prove that the proposed parametric method has the order of convergence four.In every iteration of the proposed method,only four Einstein products are required,while other iterative methods need to calculate multiple Einstein products and one tensor inversion in each iteration.Thus,the new method is superior in terms of efficiency index.Finally,the numerical comparisons performed among several well-known methods,show that the proposed method is remarkably efficient and accurate.
基金supported by the Young Scientists Fund of the Natural Science Foundation of China (Grant nos.41106165, 41106159)the Chinese Polar Environment Comprehensive Investigation & Evaluation Programmes(Grant no.CHINARE2014-04-04)+1 种基金the Project of Comprehensive Evaluation of Polar Regions on Global and Regional Climate Changes (Grant no.201105019)the National Science and Technology Support Plan of China (Grant no.2012BAC19B08)
文摘The physical decomposition method separates atmospheric variables into four parts, correlating each with solar radiation, land-sea distribution, and inter-annual and seasonal internal forcing, strengthening the anomaly signal and increasing the correlation between variables. This method was applied to the reanalysis data from the National Centers for Environmental Prediction/National Center for Atmospheric Research （NCEP/NCAR）, to study the effects of Arctic factors （Arctic oscillation （AO） and Arctic polar vortex） on wintertime temperatures in the Northern Hemisphere and China. It was fotmd that AO effects on zonal average temperature disturbance could persist for 1 month. In the AO negative phase in wintertime, the temperatures are lower in the mid-high latitudes than in normal years, but higher in low latitudes. When the polar vortex area is bigger, the zonal average temperature is lower at 50N. Influenced mainly by meridional circulation enhancement, cold air flows from high to low latitudes; thus, the temperatures in Continental Europe and the North American continent exhibit an antiphase seesaw relationship. When the AO is in negative phase and the Arctic polar vortex larger, the temperature is lower in Siberia, but higher in Greenland and the Bering Strait. Influenced by westerly troughs and ridges, the polar air disperses mainly along the tracks of atmospheric activity centers. The AO index can be considered a predictor of wintertime temperature in China. When the AO is in negative phase or the Asian polar vortex is intensified, temperatures in Northeast China and Inner Mongolia are lower, because under the influence of the Siberia High and northeast cold vortex, the cold air flows southwards.
文摘In this paper, we prove that the norm closure of all linear combinations of two unitary operators is equal to the norm closure of all invertible operators in B(H). We apply the results to frame representations and give some simple and alternative proofis of the propositions in 'P. G. Casazza, Every frame is a sum of three (but not two) orthonormal bases-and other frame representations, J. Fourier Anal. Appl., 4(6)(1998), 727-732.'
文摘IN ref. [1] for the subspaces of Hilbert space H the concept of equivalence and a generalizeddimension are introduced. Note that in refs. [2, 3] many results are true only in the separablespace. We introduce a concept of double unitary equivalence from the singular decomposition ofmatrix. In this note we get the conditions of doub1e unitary equivalent operators. Some gener-al results are obtained. It is well known that two operators A and B in Hilbert space H are called unitary equiva-lence if there is an unitary operator U such that A = U~* BU. In this case A and B can beconsidered as two complete same operators. Therefore, the condition of two-operator uitaryequivalence is very high. In the matrix theory there is a singular
文摘Let A be an m×n complex matrix. A decomposition A=QH is termed a polar decomposition of A if Q *Q=I and H is a positive semidefinite Hermitian matrix. Perturbation bounds for Q when A and its perturbation are of non-full rank matrices are provided.
基金Science Foundation of Minisitry of Education of China (No.208081)
文摘Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No.10671077)the Natural Science Foundation of Guangdong Province (Grant Nos.06025061,031496)the Research Grant Council of the Hong Kong Special Administrative Region,China (Project No.CityU 102204)
文摘In this paper,we study the perturbation bounds for the polar decomposition A=QH where Q is unitary and H is Hermitian.The optimal (asymptotic) bounds obtained in previous works for the unitary factor,the Hermitian factor and singular values of A areσ_r^2||ΔQ||_F^2≤||ΔA||_F^2, 1/2||ΔH||_F^2≤||ΔA||_F^2 and ||Δ∑||_F^2≤||ΔA||_F^2,respectively,where∑=diag(σ_1,σ_2,...,σ_r,0,...,0) is the singular value matrix of A andσ_r denotes the smallest nonzero singular value.Here we present some new combined (asymptotic) perturbation boundsσ_r^2||ΔQ||_F^2+1/2||ΔH||_F^2≤||ΔA||_F^2 andσ_r^2||ΔQ||_F^2+||Δ∑||_F^2≤||ΔA||_F^2 which are optimal for each factor.Some corresponding absolute perturbation bounds are also given.
