Take the calculation of A+ as an example. If a is a nonzero matrix of order m×n with rank r, and it is recorded as (Figure 6), and there is full rank decomposition A = f g, where (Figure 7), then (Figure 8), that is, the calculation of generalized inverse matrix is transformed into the calculation of ordinary inverse matrix. LU decomposition and QR decomposition are commonly used to achieve full rank decomposition, and then A+ is obtained. If a has singular value decomposition A=UDV*, where u and v are unitary matrices of order m and n, (Figure 9) is a matrix of order m×n, ∑ is a diagonal matrix of order r, and diagonal elements (Figure 10) are r nonzero singular values of a (square root of nonzero eigenvalue of A *), then A+=VD+U*, where You can also use Haushold transform to transform A into upper dual diagonal matrix J0=P*AQ, and then use QR algorithm to transform J0 into matrix D=G*J0h, so A=(PG)D(Qh)*, so A+ 1=(Qh)D+(PG)*. Let λ 1 be the largest non-zero eigenvalue of AA*, if 0
Grevil recursive method is also a common method to calculate A+. Let the k-th column of A be αk(k= 1, 2, …, n), A 1 = α 1, AK = (AK- 1, α k) (k = 2, 3, …, n), then
After 1955, a large number of literatures about the theory, application and calculation method of generalized inverse matrix appeared. In 1970s, some monographs and papers were published, pointing out the application of generalized inverse matrix in cybernetics, system identification, planning theory, network theory, measurement, statistics and econometrics.