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 > Diagonal Matrix example
11. Diagonal Matrix 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
2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
(Previous example)		4. Example `[[2,3],[4,10]]`
(Next example)

3. Example `[[1,1,1],[-1,-3,-3],[2,4,4]]`


Find Matrix Diagonalization ...
`[[1,1,1],[-1,-3,-3],[2,4,4]]`

Solution:
A can be diagonalized if there exists an invertible matrix P and diagonal matrix D such that `A=PDP^-1`


Here `A` = 
`1``1``1`
`-1``-3``-3`
`2``4``4`



Find eigenvalues of the matrix `A`

`|A-lamdaI|=0`

 `(1-lamda)`  `1`  `1` 
 `-1`  `(-3-lamda)`  `-3` 
 `2`  `4`  `(4-lamda)` 
 = 0


`:.(1-lamda)((-3-lamda) × (4-lamda) - (-3) × 4)-1((-1) × (4-lamda) - (-3) × 2)+1((-1) × 4 - (-3-lamda) × 2)=0`

`:.(1-lamda)((-12-lamda+lamda^2)-(-12))-1((-4+lamda)-(-6))+1((-4)-(-6-2lamda))=0`

`:.(1-lamda)(-lamda+lamda^2)-1(2+lamda)+1(2+2lamda)=0`

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

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

`:.-lamda^2(lamda-2)=0`

`:.lamda^2=0 or(lamda-2)=0 `

`:.` The eigenvalues of the matrix `A` are given by `lamda=0,2`,


1. Eigenvectors for `lamda=0`


1. Eigenvectors for `lamda=0`

`A-lamdaI = `
111
-1-3-3
244
 - `0` 
100
010
001


 = 
`1``1``1`
`-1``-3``-3`
`2``4``4`


Now, reduce this matrix
interchanging rows `R_1 harr R_3`

 = 
`2``4``4`
`-1``-3``-3`
`1``1``1`


`R_1 larr R_1-:2`

 = 
`1``2``2`
`-1``-3``-3`
`1``1``1`


`R_2 larr R_2+ R_1`

 = 
`1``2``2`
`0``-1``-1`
`1``1``1`


`R_3 larr R_3- R_1`

 = 
`1``2``2`
`0``-1``-1`
`0``-1``-1`


`R_2 larr R_2-:-1`

 = 
`1``2``2`
`0``1``1`
`0``-1``-1`


`R_1 larr R_1-2xx R_2`

 = 
`1``0``0`
`0``1``1`
`0``-1``-1`


`R_3 larr R_3+ R_2`

 = 
`1``0``0`
`0``1``1`
`0``0``0`


The system associated with the eigenvalue `lamda=0`

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


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

`=>x_1=0,x_2=-x_3`

`:.` eigenvectors corresponding to the eigenvalue `lamda=0` is

`v=`
`0`
`-x_3`
`x_3`


Let `x_3=1`

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


3. Eigenvectors for `lamda=2`



3. Eigenvectors for `lamda=2`

`A-lamdaI = `
111
-1-3-3
244
 - `2` 
100
010
001


 = 
111
-1-3-3
244
 - 
200
020
002

 = 
`-1``1``1`
`-1``-5``-3`
`2``4``2`


Now, reduce this matrix
interchanging rows `R_1 harr R_3`

 = 
`2``4``2`
`-1``-5``-3`
`-1``1``1`


`R_1 larr R_1-:2`

 = 
`1``2``1`
`-1``-5``-3`
`-1``1``1`


`R_2 larr R_2+ R_1`

 = 
`1``2``1`
`0``-3``-2`
`-1``1``1`


`R_3 larr R_3+ R_1`

 = 
`1``2``1`
`0``-3``-2`
`0``3``2`


`R_2 larr R_2-:-3`

 = 
`1``2``1`
`0``1``0.66666667`
`0``3``2`


`R_1 larr R_1-2xx R_2`

 = 
`1``0``-0.33333333`
`0``1``0.66666667`
`0``3``2`


`R_3 larr R_3-3xx R_2`

 = 
`1``0``-0.33333333`
`0``1``0.66666667`
`0``0``0`


The system associated with the eigenvalue `lamda=2`

`(A-2I)`
`x_1`
`x_2`
`x_3`
 = 
`1``0``-0.33333333`
`0``1``0.66666667`
`0``0``0`
 
`x_1`
`x_2`
`x_3`
 = 
`0`
`0`
`0`


`=>x_1-0.33333333x_3=0,x_2+0.66666667x_3=0`

`=>x_1=0.33333333x_3,x_2=-0.66666667x_3`

`:.` eigenvectors corresponding to the eigenvalue `lamda=2` is

`v=`
`0.33333333x_3`
`-0.66666667x_3`
`x_3`


Let `x_3=1`

`v_2=`
`0.33333333`
`-0.66666667`
`1`
`v_2=`
`0.33333333`
`-0.66666667`
`1`



To allow diagonalization, the number of eigenvectors must equal the matrix dimentions.
There are 2 eigenvectors which is less then 3 and therefore the matix cannot be diagonalized.
= not diagonalizable

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


2. Example `[[3,2,4],[2,0,2],[4,2,3]]`
(Previous example)		4. Example `[[2,3],[4,10]]`
(Next example)


Share this solution or page with your friends.


Home What's new College Algebra Games Feedback About us
 
Copyright © 2024. All rights reserved. Terms, Privacy
 
 

.