1, the matrix must be square, that is, the number of rows equals the number of columns.

2. The characteristic polynomial of a matrix must be completely decomposed into the product of linear factors, that is, the characteristic polynomial has no repeated characteristic roots.

3. The geometric multiplicity of each characteristic root of a matrix (the number of characteristic vectors corresponding to the characteristic root) must be equal to its algebraic multiplicity (the number of times that the corresponding characteristic root appears in the characteristic polynomial).

4. The eigenvector set corresponding to each eigenvalue of a matrix must be able to expand the entire vector space, that is, the eigenvectors are linearly independent.

If a matrix satisfies the above conditions, it can be diagonalized by similar transformation, that is, it can be expressed as the product of a diagonal matrix and a similar transformation matrix. In a diagonal matrix, the eigenvalues of the matrix are arranged on the diagonal in the order of the corresponding eigenvectors, and the elements in other positions are all zero.

The Function of Matrix in Mathematics and Science

1. Solution of linear equations: Linear equations can be expressed by matrix, and the solution of linear equations can be solved by matrix operation, such as Gaussian elimination or matrix inversion.

2. Transformation of vector space: Matrix can be used to describe linear transformation between vector spaces. For example, in computer graphics, matrices can represent translation, rotation, scaling and other transformation operations.

3. Eigenvalues and eigenvectors: Eigenvalues and eigenvectors of matrices are of great significance for understanding the properties and transformations of matrices. They can be used to describe the symmetry, stability and other important characteristics of matrices.

4. Least square method: The matrix can be used to solve the least square method. Least square method is a mathematical optimization method, which is used to solve the approximate solution of overdetermined equation.

5. Data analysis and statistics: Matrix plays an important role in data analysis and statistics. For example, covariance matrix can be used to analyze and measure the correlation between variables.

6. Graph theory and network analysis: Matrix can be used to represent graph and network structure. Adjacency matrix and incidence matrix are commonly used matrix representations to describe graphs and networks.