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Solution
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Solution provided by AtoZmath.com
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Matrix Eigenvectors (Eigenspace) calculator
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 1. `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
2. `[[3,2,4],[2,0,2],[4,2,3]]`
3. `[[1,1,1],[-1,-3,-3],[2,4,4]]`
4. `[[1,2,3],[0,1,0],[2,3,1]]`
5. `[[1,2,1],[6,-1,0],[-1,-2,-1]]`
6. `[[6,-2,2],[-2,3,-1],[2,-1,3]]`
7. `[[2,3],[4,10]]`
8. `[[5,1],[4,2]]`
9. `[[6,3],[4,5]]`
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Example1. Find Matrix Eigenvectors (Eigenspace) ...
`[[8,-6,2],[-6,7,-4],[2,-4,3]]`Solution:`|A-lamdaI|=0` | `(8-lamda)` | `-6` | `2` | | | `-6` | `(7-lamda)` | `-4` | | | `2` | `-4` | `(3-lamda)` | |
| = 0 |
`:.(8-lamda)((7-lamda) × (3-lamda) - (-4) × (-4))-(-6)((-6) × (3-lamda) - (-4) × 2)+2((-6) × (-4) - (7-lamda) × 2)=0` `:.(8-lamda)((21-10lamda+lamda^2)-16)+6((-18+6lamda)-(-8))+2(24-(14-2lamda))=0` `:.(8-lamda)(5-10lamda+lamda^2)+6(-10+6lamda)+2(10+2lamda)=0` `:. (40-85lamda+18lamda^2-lamda^3)+(-60+36lamda)+(20+4lamda)=0` `:.(-lamda^3+18lamda^2-45lamda)=0` `:.-lamda(lamda-3)(lamda-15)=0` `:.lamda=0 or (lamda-3)=0 or (lamda-15)=0` `:.lamda=0 or lamda=3 or lamda=15` `:.` The eigenvalues of the matrix `A` are given by `lamda=0,3,15` 1. Eigenvectors for `lamda=0`
1. Eigenvectors for `lamda=0` | = | | `8` | `-6` | `2` | | | `-6` | `7` | `-4` | | | `2` | `-4` | `3` | |
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Now, reduce this matrix `R_1 larr R_1-:8` | = | | `1` | `-0.75` | `0.25` | | | `-6` | `7` | `-4` | | | `2` | `-4` | `3` | |
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`R_2 larr R_2+6xx R_1` | = | | `1` | `-0.75` | `0.25` | | | `0` | `2.5` | `-2.5` | | | `2` | `-4` | `3` | |
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`R_3 larr R_3-2xx R_1` | = | | `1` | `-0.75` | `0.25` | | | `0` | `2.5` | `-2.5` | | | `0` | `-2.5` | `2.5` | |
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`R_2 larr R_2-:2.5` | = | | `1` | `-0.75` | `0.25` | | | `0` | `1` | `-1` | | | `0` | `-2.5` | `2.5` | |
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`R_1 larr R_1+0.75xx R_2` | = | | `1` | `0` | `-0.5` | | | `0` | `1` | `-1` | | | `0` | `-2.5` | `2.5` | |
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`R_3 larr R_3+2.5xx R_2` | = | | `1` | `0` | `-0.5` | | | `0` | `1` | `-1` | | | `0` | `0` | `0` | |
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The system associated with the eigenvalue `lamda=0` | `(A-0I)` | | = | | `1` | `0` | `-0.5` | | | `0` | `1` | `-1` | | | `0` | `0` | `0` | |
| | | = | |
`=>x_1-0.5x_3=0,x_2-x_3=0` `=>x_1=0.5x_3,x_2=x_3` `:.` eigenvectors corresponding to the eigenvalue `lamda=0` is Let `x_3=1` 2. Eigenvectors for `lamda=3`
2. Eigenvectors for `lamda=3` | = | | `5` | `-6` | `2` | | | `-6` | `4` | `-4` | | | `2` | `-4` | `0` | |
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Now, reduce this matrix `R_1 larr R_1-:5` | = | | `1` | `-1.2` | `0.4` | | | `-6` | `4` | `-4` | | | `2` | `-4` | `0` | |
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`R_2 larr R_2+6xx R_1` | = | | `1` | `-1.2` | `0.4` | | | `0` | `-3.2` | `-1.6` | | | `2` | `-4` | `0` | |
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`R_3 larr R_3-2xx R_1` | = | | `1` | `-1.2` | `0.4` | | | `0` | `-3.2` | `-1.6` | | | `0` | `-1.6` | `-0.8` | |
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`R_2 larr R_2-:(-3.2)` | = | | `1` | `-1.2` | `0.4` | | | `0` | `1` | `0.5` | | | `0` | `-1.6` | `-0.8` | |
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`R_1 larr R_1+1.2xx R_2` | = | | `1` | `0` | `1` | | | `0` | `1` | `0.5` | | | `0` | `-1.6` | `-0.8` | |
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`R_3 larr R_3+1.6xx R_2` | = | | `1` | `0` | `1` | | | `0` | `1` | `0.5` | | | `0` | `0` | `0` | |
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The system associated with the eigenvalue `lamda=3` | `(A-3I)` | | = | | `1` | `0` | `1` | | | `0` | `1` | `0.5` | | | `0` | `0` | `0` | |
| | | = | |
`=>x_1+x_3=0,x_2+0.5x_3=0` `=>x_1=-x_3,x_2=-0.5x_3` `:.` eigenvectors corresponding to the eigenvalue `lamda=3` is Let `x_3=1` 3. Eigenvectors for `lamda=15`
3. Eigenvectors for `lamda=15` | = | | `-7` | `-6` | `2` | | | `-6` | `-8` | `-4` | | | `2` | `-4` | `-12` | |
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Now, reduce this matrix `R_1 larr R_1-:(-7)` | = | | `1` | `0.8571` | `-0.2857` | | | `-6` | `-8` | `-4` | | | `2` | `-4` | `-12` | |
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`R_2 larr R_2+6xx R_1` | = | | `1` | `0.8571` | `-0.2857` | | | `0` | `-2.8571` | `-5.7143` | | | `2` | `-4` | `-12` | |
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`R_3 larr R_3-2xx R_1` | = | | `1` | `0.8571` | `-0.2857` | | | `0` | `-2.8571` | `-5.7143` | | | `0` | `-5.7143` | `-11.4286` | |
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`R_2 larr R_2-:(-2.8571)` | = | | `1` | `0.8571` | `-0.2857` | | | `0` | `1` | `2` | | | `0` | `-5.7143` | `-11.4286` | |
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`R_1 larr R_1-0.8571xx R_2` | = | | `1` | `0` | `-2` | | | `0` | `1` | `2` | | | `0` | `-5.7143` | `-11.4286` | |
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`R_3 larr R_3+5.7143xx R_2` | = | | `1` | `0` | `-2` | | | `0` | `1` | `2` | | | `0` | `0` | `0` | |
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The system associated with the eigenvalue `lamda=15` | `(A-15I)` | | = | | `1` | `0` | `-2` | | | `0` | `1` | `2` | | | `0` | `0` | `0` | |
| | | = | |
`=>x_1-2x_3=0,x_2+2x_3=0` `=>x_1=2x_3,x_2=-2x_3` `:.` eigenvectors corresponding to the eigenvalue `lamda=15` is Let `x_3=1` |
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