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Home > Matrix & Vector > Matrix operation > Moore-Penrose Pseudoinverse of a Matrix example
18. Moore-Penrose Pseudoinverse of a Matrix example ( Enter your problem )
( Enter your problem )
  1. Example `[[4,0],[3,-5]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
  2. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
  3. Example `[[4,0],[3,-5]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
  4. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
  5. Example `[[1,-2,3],[5,8,-1],[2,1,1],[-1,4,-3]]` `("Formula " A^(+)=(A^T*A)^(-1) * A^T)`

4. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`


Find Moore-Penrose Pseudoinverse ...
`[[1,0,1,0],[0,1,0,1]]`

Solution:
Pseudoinverse of a matrix A is `A^(+) = A^T * (A*A^T)^(-1)`


1. Find `A'`

`A^T` = 
`1``0``1``0`
`0``1``0``1`
T
 = 
`1``0`
`0``1`
`1``0`
`0``1`


2. Find `A*A'`

`A×A'`=
`1``0``1``0`
`0``1``0``1`
×
`1``0`
`0``1`
`1``0`
`0``1`


=
`1×1+0×0+1×1+0×0``1×0+0×1+1×0+0×1`
`0×1+1×0+0×1+1×0``0×0+1×1+0×0+1×1`


=
`1+0+1+0``0+0+0+0`
`0+0+0+0``0+1+0+1`


=
`2``0`
`0``2`


3. Find the inverse matrix `(A*A')^(-1)`

`|A*A'|` = 
 `2`  `0` 
 `0`  `2` 


`=2 × 2 - 0 × 0`

`=4 +0`

`=4`


`Adj(A*A')` = 
Adj
`2``0`
`0``2`


 = 
`+(2)``-(0)`
`-(0)``+(2)`
T


 = 
`2``0`
`0``2`
T


 = 
`2``0`
`0``2`


`"Now, "A*A'^(-1)=1/|A*A'| × Adj(A*A')`

 = `1/(4)` ×
`2``0`
`0``2`


 = 
`1/2``0`
`0``1/2`


4. Find the inverse matrix `A'*(A*A')^(-1)`

`A'×((A*A')^-1)`=
`1``0`
`0``1`
`1``0`
`0``1`
×
`1/2``0`
`0``1/2`


=
`1×1/2+0×0``1×0+0×1/2`
`0×1/2+1×0``0×0+1×1/2`
`1×1/2+0×0``1×0+0×1/2`
`0×1/2+1×0``0×0+1×1/2`


=
`1/2+0``0+0`
`0+0``0+1/2`
`1/2+0``0+0`
`0+0``0+1/2`


=
`1/2``0`
`0``1/2`
`1/2``0`
`0``1/2`


`:.` Moore-Penrose pseudoinverse `A^(+)=`
`1/2``0`
`0``1/2`
`1/2``0`
`0``1/2`



This material is intended as a summary. Use your textbook for detail explanation.
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