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Home > Matrix & Vector calculators > Moore-Penrose Pseudoinverse of a Matrix example
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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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Other related methods
- Transforming matrix to Row Echelon Form
- Transforming matrix to Reduced Row Echelon Form
- Rank of matrix
- Characteristic polynomial of matrix
- Eigenvalues
- Eigenvectors (Eigenspace)
- Triangular Matrix
- LU decomposition using Gauss Elimination method of matrix
- LU decomposition using Doolittle's method of matrix
- LU decomposition using Crout's method of matrix
- Diagonal Matrix
- Cholesky Decomposition
- QR Decomposition (Gram Schmidt Method)
- QR Decomposition (Householder Method)
- LQ Decomposition
- Pivots
- Singular Value Decomposition (SVD)
- Moore-Penrose Pseudoinverse
- Power Method for dominant eigenvalue
- determinants using Sarrus Rule
- determinants using properties of determinants
- Row Space
- Column Space
- Null Space
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2. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
(Previous example) | 4. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
(Next example) |
3. Example `[[4,0],[3,-5]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
Find Moore-Penrose Pseudoinverse of a Matrix ...
`[[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')`
| = | | `0.085` | `-0.03` | | | `-0.03` | `0.04` | |
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4. Find the inverse matrix `A'*(A*A')^(-1)`
| `A'×((A*A')^-1)` | = | | × | | `0.085` | `-0.03` | | | `-0.03` | `0.04` | |
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| = | | `4×0.085+3×-0.03` | `4×-0.03+3×0.04` | | | `0×0.085-5×-0.03` | `0×-0.03-5×0.04` | |
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| = | | `0.34-0.09` | `-0.12+0.12` | | | `0+0.15` | `0-0.2` | |
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| `:.` Moore-Penrose pseudoinverse `A^(+)=` | |
This material is intended as a summary. Use your textbook for detail explanation.
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2. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=V Sigma^(+) U^T)`
(Previous example) | 4. Example `[[1,0,1,0],[0,1,0,1]]` `("Formula " A^(+)=A^T * (A*A^T)^(-1))`
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