This is because A is full rank, then r(A)=r(A|b).

If m=n at this time, there is a unique solution.

M<n has infinite solutions.

M>n is impossible because the rank of a matrix is equal to the rank of rows and columns, but it cannot exceed the number of rows or columns. At this time, r(A)=m > appears; The number of columns is n, so it is impossible.

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 extended data matrix

Main project: matrix decomposition

Matrix decomposition is to decompose a matrix into the sum or product of several relatively simple or characteristic matrices. Matrix decomposition methods generally include triangular decomposition, spectral decomposition, singular value decomposition, full rank decomposition and so on.

spectroanalysis

Spectral decomposition is a method to decompose a matrix into the product of its eigenvalue and eigenvector. It should be noted that only diagonalizable matrices can be decomposed into features.

singular value decomposition SVD

Suppose m is an m×n matrix, in which all elements belong to the field k, that is, the real number field or the complex number field. So this decomposition makes

Where u is m×m unitary matrix; σ is a real diagonal matrix of order m×n; And V*, that is, the * * * yoke transposition of v, is a unitary matrix of order n × n, and this decomposition is called singular value decomposition of m [19]. The element σ i on the diagonal of σ is the singular value of m, and the usual practice is to arrange the singular values from large to small. So σ can be uniquely determined by m.