1. is Orthogonal Matrix ?
`[[1,0],[0,-1]]`
Solution:
A square matrix `A`, such that `A xx A^T = I`, is called an orthogonal matrix, where `I` is an identity matrix and `A^T` is the transpose of matrix `A`.
| = | | `1×1+0×0` | `1×0+0×-1` | | | `0×1-1×0` | `0×0-1×-1` | |
|
`A xx A^T = I`, So `A` is an orthogonal matrix
2. is Orthogonal Matrix ?
`[[1,2,3],[4,5,6],[7,8,9]]`
Solution:
A square matrix `A`, such that `A xx A^T = I`, is called an orthogonal matrix, where `I` is an identity matrix and `A^T` is the transpose of matrix `A`.
| `A` | = | | `1` | `2` | `3` | | | `4` | `5` | `6` | | | `7` | `8` | `9` | |
|
| `A^T` | = | | `1` | `2` | `3` | | | `4` | `5` | `6` | | | `7` | `8` | `9` | |
| T |
| = | | `1` | `4` | `7` | | | `2` | `5` | `8` | | | `3` | `6` | `9` | |
|
| `A×(A^T)` | = | | `1` | `2` | `3` | | | `4` | `5` | `6` | | | `7` | `8` | `9` | |
| × | | `1` | `4` | `7` | | | `2` | `5` | `8` | | | `3` | `6` | `9` | |
|
| = | | `1×1+2×2+3×3` | `1×4+2×5+3×6` | `1×7+2×8+3×9` | | | `4×1+5×2+6×3` | `4×4+5×5+6×6` | `4×7+5×8+6×9` | | | `7×1+8×2+9×3` | `7×4+8×5+9×6` | `7×7+8×8+9×9` | |
|
| = | | `1+4+9` | `4+10+18` | `7+16+27` | | | `4+10+18` | `16+25+36` | `28+40+54` | | | `7+16+27` | `28+40+54` | `49+64+81` | |
|
| = | | `14` | `32` | `50` | | | `32` | `77` | `122` | | | `50` | `122` | `194` | |
|
`A xx A^T != I`, So `A` is not an orthogonal matrix
This material is intended as a summary. Use your textbook for detail explanation.
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