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Find Pseudoinverse [[4,0],[3,-5]] [ Calculator, Method and examples ]

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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`


`A = `
`4``0`
`3``-5`


`A * A'`
`A^T` = 
`4``0`
`3``-5`
T
 = 
`4``3`
`0``-5`


`A×(A^T)`=
`4``0`
`3``-5`
×
`4``3`
`0``-5`


=
`4×4+0×0``4×3+0×-5`
`3×4-5×0``3×3-5×-5`


=
`16+0``12+0`
`12+0``9+25`


=
`16``12`
`12``34`
`A * A' = `
`16``12`
`12``34`


Find Eigen vector for `A * A'`

`|A * A'-lamdaI|=0`

 `(16-lamda)`  `12` 
 `12`  `(34-lamda)` 
 = 0


`:.(16-lamda) × (34-lamda) - 12 × 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 `

`:.` 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 = `
1612
1234
 - `40` 
10
01


 = 
1612
1234
 - 
400
040

 = 
`-24``12`
`12``-6`


Now, reduce this matrix
`R_1 larr R_1-:-24`

 = 
 `1` `1=-24-:-24`
`R_1 larr R_1-:-24`
 `-0.5` `-0.5=12-:-24`
`R_1 larr R_1-:-24`
`12``-6`


`R_2 larr R_2-12xx R_1`

 = 
`1``-0.5`
 `0` `0=12-12xx1`
`R_2 larr R_2-12xx R_1`
 `0` `0=-6-12xx-0.5`
`R_2 larr R_2-12xx R_1`


The system associated with the eigenvalue `lamda=40`

`(A * A'-40I)`
`x_1`
`x_2`
 = 
`1``-0.5`
`0``0`
 
`x_1`
`x_2`
 = 
`0`
`0`


`=>x_1-0.5x_2=0`

`=>x_1=0.5x_2`

`:.` eigenvectors corresponding to the eigenvalue `lamda=40` is

`v=`
`0.5x_2`
`x_2`


Let `x_2=1`

`v_1=`
`0.5`
`1`
`v_1=`
`0.5`
`1`


2. Eigenvectors for `lamda=10`




2. Eigenvectors for `lamda=10`

`A * A'-lamdaI = `
1612
1234
 - `10` 
10
01


 = 
1612
1234
 - 
100
010

 = 
`6``12`
`12``24`


Now, reduce this matrix





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