The Uncertainty Principle And The Geometric Mean Of The Matrix Manifold

In this thesis,we obtain the Heisenberg uncertainty principle for the n dimension linear canonical transformation,and using the definition of the geometric mean of the symmetric positive definite matrices manifold,we generalize the entropy uncertainty principle.The contents in this thesis are summarized as the following.1.The linear canonical transformation is the generalization of the Fourier transfor-mation,and it has very important applications in mathematics,physics and information theory.In this thesis,by virtue of the definition of linear canonical transformation and the Heisenberg uncertainty principle for Fourier transformation,we obtain the Heisen-berg uncertainty principle for n dimension linear canonical transformation based on the properties of the integral transform.Also we get one uncertainty principle for the Wigner Ville distribution(WVD)by using the Heisenberg uncertainty principle of the linear canonical transformation to Wigner-Ville distribution.Furthermore the Heisen-berg uncertainty principle for linear canonical transformation implies this uncertainty principle is only related to the matrix B.2.Since the linear canonical transformation is a special kind of the linear operator which satisfies the Hausdorff-Young inequality,by using the definitions of the density for the class trace operator and its Rumin conjugate and also using the Riesz-Thorin theorem,we obtain the Tsallis entropy uncertainty principle for these linear operator.Since the Tsallis entropy uncertainty principle is the generalization of the Shannon entropy uncertainty principle and the Heisenberg uncertainty principle,in the certain conditions,the Tsallis entropy uncertainty principle can imply the Shannon entropy uncertainty principle and the Heisenberg uncertainty principle.3.All the symmetric positive definite matrices can become a Riemannian manifold with the non-positive sectional curvature,and in this Riemannian manifold,the mid-point of the geodesic for any two symmetric positive definite matrices is defined as the geometric mean.In this thesis,by means of the Peierls-Bogoliubov inequality for the symmetric positive definite matrix and the definitions of the class trace operator and its Rumin conjugate,we obtain the uncertainty principle of the Renyi entropy.Then by using the Baker-Campbell-Hausdorff formula and the definition of the geometric mean for the symmetric positive definite matrix,we obtain the entropy uncertainty principle for the class trace operator and its Rumin conjugate,which is the generalization of the Shannon entropy uncertainty principle.4.Since the Shannon entropy is defined in the Euclidean space which is one of the geodesic complete Riemannian manifold,in this thesis,we generalize the definition of the Shannon entropy to the geodesic complete Riemannian manifold,and we obtain one important inequality for the Shannon entropy.Then we obtain one inequality between the Shannon entropy in the symmetric positive definite matrices manifold and the Shannon entropy in the Euclidean space by the monotonicity of matrix function.