Problem: Pseudoinverse [[4,0],[3,-5]] [ Calculator, Method and examples ]
Solution:
Your problem `->` Pseudoinverse [[4,0],[3,-5]]
The Moore-Penrose pseudoinverse `A^(+)` is calculated from SVD (Singular Value Decomposition) of a matrix A,
`A = U Sigma V^T`
then Moore-Penrose pseudoinverse `A^(+)` is given by
`A^(+) = V Sigma^(+) U^T`
where `Sigma^(+)` is obtained by taking the reciprocal of each non-zero data on the diagonal of `Sigma`, leaving all other zeros as it is, and then taking transpose of the resultant matrix.
`U, Sigma, V` using SVD : `A = U Sigma V^T`
Here we are trying to find out two solutions
using `A*A'` and `A'*A`
`1^"st"` Solution using `A*A'` for normalized vectors `u_i``A * A'`
| = | | `4 xx 4+0 xx 0` | `4 xx 3+0 xx (-5)` | | | `3 xx 4+(-5) xx 0` | `3 xx 3+(-5) xx (-5)` | |
|
Find Eigen vector for `A * A'`
`|A * A'-lamdaI|=0`
| `(16-lamda)` | `12` | | | `12` | `(34-lamda)` | |
| = 0 |
`:.(16-lamda) xx (34-lamda) - 12 xx 12=0`
`:.(544-50lamda+lamda^2)-144=0`
`:.(lamda^2-50lamda+400)=0`
`:.(lamda-10)(lamda-40)=0`
`:.(lamda-10)=0 or (lamda-40)=0`
`:.lamda=10 or lamda=40`
`:.` The eigenvalues of the matrix `A * A'` are given by `lamda=10,40`
1. Eigenvectors for `lamda=40`
1. Eigenvectors for `lamda=40`
| `A * A'-lamdaI = ` | | - `40` | |
Now, reduce this matrix
`R_1 "(new)"=R_1 "(old)" -:(-24)`
| `R_1"(old)" = ` | `-24` | `12` |
| `R_1 "(new)"=R_1 "(old)" -:(-24)` | `1` | `-0.5` |
`R_2 "(new)"=R_2 "(old)" -12xx R_1 "(old)"`
| `R_2"(old)" = ` | `12` | `-6` |
| `R_1"(old)" = ` | `1` | `-0.5` |
| `12xx R_1"(old)" = ` | `12` | `-6` |
| `R_2 "(new)"=R_2 "(old)" -12xx R_1 "(old)"` | `0` | `0` |
The system associated with the eigenvalue `lamda=40`
`=>x_1-0.5x_2=0`
`=>x_1=0.5x_2`
`:.` eigenvectors corresponding to the eigenvalue `lamda=40` is
Let `x_2=1`
2. Eigenvectors for `lamda=10`
2. Eigenvectors for `lamda=10`
| `A * A'-lamdaI = ` | | - `10` | |
