Its inverse matrix is also a diagonal matrix, and the elements on the diagonal are exactly the reciprocal of the elements on the diagonal of the corresponding original matrix.
It can be proved by the elementary transformation method of the inverse matrix, so the inverse matrix is as follows:
Extended data:
In mathematics, a matrix is a group of complex numbers or real numbers arranged in a rectangular array, which originated from a square matrix composed of coefficients and constants of equations. This concept was first put forward by British mathematician Kelly in19th century.
Matrix is a common tool in applied mathematics such as advanced algebra and statistical analysis. In physics, matrices have applications in circuit science, mechanics, optics and quantum physics. In computer science, three-dimensional animation also needs matrix. Matrix operation is an important problem in the field of numerical analysis. Decomposition of a matrix into a combination of simple matrices can simplify the operation of the matrix in theory and practical application.
For some widely used and special matrices, such as sparse matrix and quasi-diagonal matrix, there are concrete fast operation algorithms. For the development and application of matrix related theory, please refer to matrix theory. Infinite-dimensional matrices will also appear in astrophysics, quantum mechanics and other fields, which is the generalization of matrices.
The main branch of numerical analysis is devoted to the development of effective algorithms for matrix calculation, which has been a topic for centuries and an expanding research field. The matrix decomposition method simplifies the theoretical and practical calculation. The customized algorithm for specific matrix structures (such as sparse matrix and near-angle matrix) speeds up the operation speed in finite element method and other calculations. Infinite matrix appears in planetary theory and atomic theory. A simple example of infinite matrix is the matrix representing the derivative operator of Taylor series of functions.
definition
A table with m rows and n columns arranged according to m × n numbers aij is called m rows and n columns matrix, or m × n matrix for short. Write down:
This number of m×n is called the element of matrix A, which is called element for short. The number aij is located in the I-th row and the J-th column of matrix A, which is called the (I, j) element of matrix A. A matrix with the number aij as the (I, j) element can be denoted as (aij) or (aij)m × n, and the m × n matrix A is also denoted as Amn.
A matrix with real elements is called a real matrix, and a matrix with complex elements is called a complex matrix. A matrix with the number of rows and columns equal to n is called an n-order matrix or an n-order square matrix.
Reference: Baidu Encyclopedia-Matrix