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Home > Matrix & Vector calculators > Eigenvectors example
6. Eigenvectors (Eigenspace) example ( Enter your problem )
( Enter your problem )
  1. Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
  2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
  3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
  4. Example `[[2,3],[4,10]]`
 		Other related methods
  1. Transforming matrix to Row Echelon Form
  2. Transforming matrix to Reduced Row Echelon Form
  3. Rank of matrix
  4. Characteristic polynomial of matrix
  5. Eigenvalues
  6. Eigenvectors (Eigenspace)
  7. Triangular Matrix
  8. LU decomposition using Gauss Elimination method of matrix
  9. LU decomposition using Doolittle's method of matrix
  10. LU decomposition using Crout's method of matrix
  11. Diagonal Matrix
  12. Cholesky Decomposition
  13. QR Decomposition (Gram Schmidt Method)
  14. QR Decomposition (Householder Method)
  15. LQ Decomposition
  16. Pivots
  17. Singular Value Decomposition (SVD)
  18. Moore-Penrose Pseudoinverse
  19. Power Method for dominant eigenvalue
  20. determinants using Sarrus Rule
  21. determinants using properties of determinants
  22. Row Space
  23. Column Space
  24. Null Space
1. Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
(Previous example)		3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
(Next example)

2. Example `[[3,2,4],[2,0,2],[4,2,3]]`


Find Eigenvectors ...
`[[3,2,4],[2,0,2],[4,2,3]]`

Solution:
`|A-lamdaI|=0`

 `(3-lamda)`  `2`  `4` 
 `2`  `(-lamda)`  `2` 
 `4`  `2`  `(3-lamda)` 
 = 0


`:.(3-lamda)((-lamda) × (3-lamda) - 2 × 2)-2(2 × (3-lamda) - 2 × 4)+4(2 × 2 - (-lamda) × 4)=0`

`:.(3-lamda)((-3lamda+lamda^2)-4)-2((6-2lamda)-8)+4(4-(-4lamda))=0`

`:.(3-lamda)(-4-3lamda+lamda^2)-2(-2-2lamda)+4(4+4lamda)=0`

`:. (-12-5lamda+6lamda^2-lamda^3)-(-4-4lamda)+(16+16lamda)=0`

`:.(-lamda^3+6lamda^2+15lamda+8)=0`

`:.-(lamda+1)(lamda+1)(lamda-8)=0`

`:.` The eigenvalues of the matrix A are given by `lamda=-1,-1,8`


1. Eigenvectors for `lamda=-1`

`A-lamdaI = `
324
202
423
 + `1` 
100
010
001


 = 
324
202
423
 + 
100
010
001

 = 
`4``2``4`
`2``1``2`
`4``2``4`


Now, reduce this matrix
`R_1 larr R_1-:4`

 = 
 `1` `1=4-:4`
`R_1 larr R_1-:4`		 `1/2` `1/2=2-:4`
`R_1 larr R_1-:4`		 `1` `1=4-:4`
`R_1 larr R_1-:4`
`2``1``2`
`4``2``4`


`R_2 larr R_2-2xx R_1`

 = 
`1``1/2``1`
 `0` `0=2-2xx1`
`R_2 larr R_2-2xx R_1`		 `0` `0=1-2xx1/2`
`R_2 larr R_2-2xx R_1`		 `0` `0=2-2xx1`
`R_2 larr R_2-2xx R_1`
`4``2``4`


`R_3 larr R_3-4xx R_1`

 = 
`1``1/2``1`
`0``0``0`
 `0` `0=4-4xx1`
`R_3 larr R_3-4xx R_1`		 `0` `0=2-4xx1/2`
`R_3 larr R_3-4xx R_1`		 `0` `0=4-4xx1`
`R_3 larr R_3-4xx R_1`


The system associated with the eigenvalue `lamda=-1`

`(A+1I)`
`x_1`
`x_2`
`x_3`
 = 
`1``1/2``1`
`0``0``0`
`0``0``0`
 
`x_1`
`x_2`
`x_3`
 = 
`0`
`0`
`0`


`=>x_1+1/2x_2+x_3=0`

`=>x_1=-1/2x_2-x_3`

`:.` eigenvectors corresponding to the eigenvalue `lamda=-1` is

`v=`
`-1/2x_2-x_3`
`x_2`
`x_3`


Let `x_2=1,x_3=0`

`v_1=`
`-1/2`
`1`
`0`


Let `x_2=0,x_3=1`

`v_2=`
`-1`
`0`
`1`



3. Eigenvectors for `lamda=8`

`A-lamdaI = `
324
202
423
 - `8` 
100
010
001


 = 
324
202
423
 - 
800
080
008

 = 
`-5``2``4`
`2``-8``2`
`4``2``-5`


Now, reduce this matrix
`R_1 larr R_1-:-5`

 = 
 `1` `1=-5-:-5`
`R_1 larr R_1-:-5`		 `-2/5` `-2/5=2-:-5`
`R_1 larr R_1-:-5`		 `-4/5` `-4/5=4-:-5`
`R_1 larr R_1-:-5`
`2``-8``2`
`4``2``-5`


`R_2 larr R_2-2xx R_1`

 = 
`1``-2/5``-4/5`
 `0` `0=2-2xx1`
`R_2 larr R_2-2xx R_1`		 `-36/5` `-36/5=-8-2xx-2/5`
`R_2 larr R_2-2xx R_1`		 `18/5` `18/5=2-2xx-4/5`
`R_2 larr R_2-2xx R_1`
`4``2``-5`


`R_3 larr R_3-4xx R_1`

 = 
`1``-2/5``-4/5`
`0``-36/5``18/5`
 `0` `0=4-4xx1`
`R_3 larr R_3-4xx R_1`		 `18/5` `18/5=2-4xx-2/5`
`R_3 larr R_3-4xx R_1`		 `-9/5` `-9/5=-5-4xx-4/5`
`R_3 larr R_3-4xx R_1`


`R_2 larr R_2xx-5/36`

 = 
`1``-2/5``-4/5`
 `0` `0=0xx-5/36`
`R_2 larr R_2xx-5/36`		 `1` `1=-36/5xx-5/36`
`R_2 larr R_2xx-5/36`		 `-1/2` `-1/2=18/5xx-5/36`
`R_2 larr R_2xx-5/36`
`0``18/5``-9/5`


`R_1 larr R_1+2/5xx R_2`

 = 
 `1` `1=1+2/5xx0`
`R_1 larr R_1+2/5xx R_2`		 `0` `0=-2/5+2/5xx1`
`R_1 larr R_1+2/5xx R_2`		 `-1` `-1=-4/5+2/5xx-1/2`
`R_1 larr R_1+2/5xx R_2`
`0``1``-1/2`
`0``18/5``-9/5`


`R_3 larr R_3-18/5xx R_2`

 = 
`1``0``-1`
`0``1``-1/2`
 `0` `0=0-18/5xx0`
`R_3 larr R_3-18/5xx R_2`		 `0` `0=18/5-18/5xx1`
`R_3 larr R_3-18/5xx R_2`		 `0` `0=-9/5-18/5xx-1/2`
`R_3 larr R_3-18/5xx R_2`


The system associated with the eigenvalue `lamda=8`

`(A-8I)`
`x_1`
`x_2`
`x_3`
 = 
`1``0``-1`
`0``1``-1/2`
`0``0``0`
 
`x_1`
`x_2`
`x_3`
 = 
`0`
`0`
`0`


`=>x_1-x_3=0,x_2-1/2x_3=0`

`=>x_1=x_3,x_2=1/2x_3`

`:.` eigenvectors corresponding to the eigenvalue `lamda=8` is

`v=`
`x_3`
`1/2x_3`
`x_3`


Let `x_3=1`

`v_3=`
`1`
`1/2`
`1`


This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


1. Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
(Previous example)		3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`
(Next example)


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