We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
 
 
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra
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. Determinant by gaussian elimination
  21. Expanding determinant along row / column
  22. Determinants using montante (bareiss algorithm)
  23. Leibniz formula for determinant
  24. determinants using Sarrus Rule
  25. determinants using properties of determinants
  26. Row Space
  27. Column Space
  28. 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 Matrix Eigenvectors (Eigenspace) ...
`[[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`

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

`:.lamda=-1 or lamda=-1 or lamda=8`

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

1. Eigenvectors for `lamda=-1`



1. Eigenvectors for `lamda=-1`

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


 = 
324
202
423
 - 
-100
0-10
00-1

 = 
`4``2``4`
`2``1``2`
`4``2``4`


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

 = 
`1``0.5``1`
`2``1``2`
`4``2``4`


`R_2 larr R_2-2xx R_1`

 = 
`1``0.5``1`
`0``0``0`
`4``2``4`


`R_3 larr R_3-4xx R_1`

 = 
`1``0.5``1`
`0``0``0`
`0``0``0`


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

`(A-(-1)I)`
`x_1`
`x_2`
`x_3`
 = 
`1``0.5``1`
`0``0``0`
`0``0``0`
 
`x_1`
`x_2`
`x_3`
 = 
`0`
`0`
`0`


`=>x_1+0.5x_2+x_3=0`

`=>x_1=-0.5x_2-x_3`

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

`v=`
`-0.5x_2-x_3`
`x_2`
`x_3`


Let `x_2=1,x_3=0`

`v_1=`
`-0.5`
`1`
`0`


Let `x_2=0,x_3=1`

`v_2=`
`-1`
`0`
`1`
`v_1=`
`-0.5`
`1`
`0`
,`v_2=`
`-1`
`0`
`1`


3. Eigenvectors for `lamda=8`



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``-0.4``-0.8`
`2``-8``2`
`4``2``-5`


`R_2 larr R_2-2xx R_1`

 = 
`1``-0.4``-0.8`
`0``-7.2``3.6`
`4``2``-5`


`R_3 larr R_3-4xx R_1`

 = 
`1``-0.4``-0.8`
`0``-7.2``3.6`
`0``3.6``-1.8`


`R_2 larr R_2-:(-7.2)`

 = 
`1``-0.4``-0.8`
`0``1``-0.5`
`0``3.6``-1.8`


`R_1 larr R_1+0.4xx R_2`

 = 
`1``0``-1`
`0``1``-0.5`
`0``3.6``-1.8`


`R_3 larr R_3-3.6xx R_2`

 = 
`1``0``-1`
`0``1``-0.5`
`0``0``0`


The system associated with the eigenvalue `lamda=8`

`(A-8I)`
`x_1`
`x_2`
`x_3`
 = 
`1``0``-1`
`0``1``-0.5`
`0``0``0`
 
`x_1`
`x_2`
`x_3`
 = 
`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=8` is

`v=`
`x_3`
`0.5x_3`
`x_3`


Let `x_3=1`

`v_3=`
`1`
`0.5`
`1`
`v_3=`
`1`
`0.5`
`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)


Share this solution or page with your friends.
 
 
Home What's new College Algebra Games Feedback About us
 
Copyright © 2025. All rights reserved. Terms, Privacy
 
 

.