Matrix Inequalities On Frobenius Norm And Hadamard-like Determinant

As an important branch of the matrix theory, the inequalities concerning matrixnorms and determinant have been attracting many scholars’ attention. The purpose ofthis article is to research the inequalities relating to Frobenius norm and the determinantof matrices. The main results and innovations are as follows:1. New Young inequalities of the positive definite matrix for the Frobenius normare investigated. These new inequalities depict the Young inequality from new point.Compared with the classical Young inequality, our results are tighter.2. We give an improvement of Omar Hirzallah and Fuad Kittaneh’s conclusions.Furthermore, we extend them to the Hermite matrix, which are more generalresults than the before.3. We improve Omar Hirzallah’s inequalities further and Heinz inequalities forFrobenius norm are given.4. Partial proof of a problem called Hadamard-like inequality is given. Thisproblem is proposed by Minghua Lin and released by Image. For a special case oftri-diagonal matrices we prove it holds for n3.5. Finally, we give a new Hadamard-like inequality that holds for all Hermitematrices of order n (n2).