Algebraic Properties Of Quantum Matrices

This paper is a summary of the basic definition and properties of a special matrix Manin matrix in the quantum matrix. The article is based on A. Cher-vov, G. Falqui, V. Rubsov’s article Properties of Manin Algebraic Matrices, [22], [11], [5], [17]and [25] summed up the Manin matrix of a series of proper-ties, and with the 2×2 Manin matrix as an example of a detailed exposition. The main idea is the basic properties and basic definition of Manin matrix, such as Cramer’s rule, Schur formula, Jacobi ratio theorems, Cayley-Hamilton theorem, Frobenius theorem, using Leningrad symbols show the relationships between the Manin matrix.The first chapter mainly introduces the basic definition of Manin ma-trix and q-Manin matrix, as well as the definition of the determinant and permanent [7] [6], in the basic definition of the basis to draw some simple con-clusions of Manin matrix; the second chapter mainly introduces the basic prop-erties of inverse Manin matrix[7][16], a Manin matrix of the Cramer rule, the basic properties of the left and right inverse, and the 2×2 order Manin matrix as an example of a detailed proof; the third chapter mainly introduces the Schur complement theorem and Jacobi’s ratio theorem [13], [?], the multiplication of special block matrix determinant and Sylvester determinant equivalence [24]; the fourth chapter mainly introduces the Cayley-Hamilton theorem, Frobenius theorem and Gauss decomposition; the fifth chapter is using the Leningrad symbols show the tensor product forms in 2x 2 order Manin matrix [12][15].This paper maninly summarizes the basic properties and the basic defini-tion of the Manin matrix. And on this basis, using the 2×2 Manin matrix, which is the most simple Manin matrix, as an example has a detailed explaina- tion of the various definitions and properties.