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18. Moore-Penrose Pseudoinverse of a Matrix example
( Enter your problem )
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- Example `[[4,0],[3,-5]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
- Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
- Example `[[4,0],[3,-5]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
- Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
- Example `[[1,-2,3],[5,8,-1],[2,1,1],[-1,4,-3]]` `("Formula " A^(+)=(A^T*A)^(-1) * A^T)`
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3. Example `[[4,0],[3,-5]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
Find Moore-Penrose Pseudoinverse ...
`[[4,0],[3,-5]]`Solution:Pseudoinverse of a matrix A is `A^(+) = A^T * (A*A^T)^(-1)` 1. Find `A'`2. Find `A*A'`| = | | `4×4+0×0` | `4×3+0×-5` | | | `3×4-5×0` | `3×3-5×-5` | |
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3. Find the inverse matrix `(A*A')^(-1)``=16 × 34 - 12 × 12` `=544 -144` `=400` | = | | `+(34)` | `-(12)` | | | `-(12)` | `+(16)` | |
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`"Now, "A*A'^(-1)=1/|A*A'| × Adj(A*A')` | = | | `17/200` | `-3/100` | | | `-3/100` | `1/25` | |
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4. Find the inverse matrix `A'*(A*A')^(-1)`| `A'×((A*A')^-1)` | = | | × | | `17/200` | `-3/100` | | | `-3/100` | `1/25` | |
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| = | | `4×17/200+3×-3/100` | `4×-3/100+3×1/25` | | | `0×17/200-5×-3/100` | `0×-3/100-5×1/25` | |
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| = | | `17/50-9/100` | `-3/25+3/25` | | | `0+3/20` | `0-1/5` | |
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| `:.` Moore-Penrose pseudoinverse `A^(+)=` | |
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