Matrix Trace Inequality And Its Applications

Matrix Trace Inequality is one of the main subjects of matrix theory, many of volume of calculation will eventually attributed to the matrix trace operator. As a mathematics instrument, control inequalities can usually describe much internal relationship between mathematical quantities and almost penetrate to every mathematical domain and display an excellent role. Frobenius norm is a typical unitarily invariant norm; it is the primary means of researching least squares solutions and matrix perturbation. Hadamard product is a special matrix multiplication and it has the important application in combinatorial scheme of combinatorial theory, characteristic function of probability theory and communication engineering and so on.This paper is divided into6sections. First section is an overview of Lemma and symbols; Second section deals with characteristic value and singular value of complex matrix and Hermite matrix under certain conditions, control inequalities and norm inequalities were studied at the same time; world renowned Neumann inequalities has been introduced and discussed in third part; two singular value inequalities about the trace of Hadamard matrix was introduced in the following section, then we introduced the inequalities of traces of several complex matrix continually multiply product. By using characteristic values and the property of singular value, we got m complex matrix products as well as inequalities of Hemite semi definite matrix even power products in part five; at last, in section six, the application of traces of matrix in matrix approximation in under the approximation theory.