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.

Eigenvalues and eigenvectors of matrices

An eigenvalues and corresponding eigenvectors of n×n block matrix A satisfy scalar and nonzero vectors.

. Where v is the eigenvector,

Is the eigenvalue.

The sum of all eigenvalues of a is called the spectrum of a.

, remember as

. The eigenvalues and eigenvectors of the matrix can reveal the deep features of linear transformation.