1. Example `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
1. `[[8,-6,2],[-6,7,-4],[2,-4,3]]`
Find eigenvectors ...
`|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`
`:.` the eigenvalues of the matrix A are given by `lamda=0,3,15`
1. Eigenvectors for `lamda=0`
`A-lamdaI = [[8,-6,2],[-6,7,-4],[2,-4,3]]-0[[1,0,0],[0,1,0],[0,0,1]]`
`=[[8,-6,2],[-6,7,-4],[2,-4,3]]`
Now, reduce this matrix
Dividing `R_1` by `8`
`=[[1,-3/4,1/4],[-6,7,-4],[2,-4,3]]`
`R_2 larr R_2 + 6 * R1`
`=[[1,-3/4,1/4],[0,5/2,-5/2],[2,-4,3]]`
`R_3 larr R_3 - 2 * R_1`
`=[[1,-3/4,1/4],[0,5/2,-5/2],[0,-5/2,5/2]]`
Dividing `R_2` by `5/2`
`=[[1,-3/4,1/4],[0,1,-1],[0,-5/2,5/2]]`
`R_1 larr R_1 + 3/4 * R2`
`=[[1,0,-1/2],[0,1,-1],[0,-5/2,5/2]]`
`R_3 larr R_3 + 5/2 * R2`
`=[[1,0,-1/2],[0,1,-1],[0,0,0]]`
The system associated with the eigenvalue `lamda=0`
`(A-0I)[[x_1],[x_2],[x_3]]=[[1,0,-1/2],[0,1,-1],[0,0,0]][[x_1],[x_2],[x_3]]=[[0],[0],[0]]`
`i.e.`
`x_1-1/2x_3=0`
`x_2-x_3=0`
`i.e.`
`x_1=1/2x_3`
`x_2=x_3`
`:.` eigenvector corresponding to the eigenvalue `lamda=0` is
`v=[[1/2x_3],[x_3],[x_3]]`
Let `x_3=1`
`v_1=[[1/2],[1],[1]]`
2. Eigenvectors for `lamda=3`
`A-lamdaI = [[8,-6,2],[-6,7,-4],[2,-4,3]]-3[[1,0,0],[0,1,0],[0,0,1]]`
`=[[8,-6,2],[-6,7,-4],[2,-4,3]]-[[3,0,0],[0,3,0],[0,0,3]]`
`=[[5,-6,2],[-6,4,-4],[2,-4,0]]`
Now, reduce this matrix
Dividing `R_1` by `5`
`=[[1,-6/5,2/5],[-6,4,-4],[2,-4,0]]`
`R_2 larr R_2 + 6 * R1`
`=[[1,-6/5,2/5],[0,-16/5,-8/5],[2,-4,0]]`
`R_3 larr R_3 - 2 * R_1`
`=[[1,-6/5,2/5],[0,-16/5,-8/5],[0,-8/5,-4/5]]`
Dividing `R_2` by `-16/5`
`=[[1,-6/5,2/5],[0,1,1/2],[0,-8/5,-4/5]]`
`R_1 larr R_1 + 6/5 * R2`
`=[[1,0,1],[0,1,1/2],[0,-8/5,-4/5]]`
`R_3 larr R_3 + 8/5 * R2`
`=[[1,0,1],[0,1,1/2],[0,0,0]]`
The system associated with the eigenvalue `lamda=3`
`(A-3I)[[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]]`
`i.e.`
`x_1+x_3=0`
`x_2+1/2x_3=0`
`i.e.`
`x_1=-x_3`
`x_2=-1/2x_3`
`:.` eigenvector corresponding to the eigenvalue `lamda=3` is
`v=[[-x_3],[-1/2x_3],[x_3]]`
Let `x_3=1`
`v_2=[[-1],[-1/2],[1]]`
3. Eigenvectors for `lamda=15`
`A-lamdaI = [[8,-6,2],[-6,7,-4],[2,-4,3]]-15[[1,0,0],[0,1,0],[0,0,1]]`
`=[[8,-6,2],[-6,7,-4],[2,-4,3]]-[[15,0,0],[0,15,0],[0,0,15]]`
`=[[-7,-6,2],[-6,-8,-4],[2,-4,-12]]`
Now, reduce this matrix
Dividing `R_1` by `-7`
`=[[1,6/7,-2/7],[-6,-8,-4],[2,-4,-12]]`
`R_2 larr R_2 + 6 * R1`
`=[[1,6/7,-2/7],[0,-20/7,-40/7],[2,-4,-12]]`
`R_3 larr R_3 - 2 * R_1`
`=[[1,6/7,-2/7],[0,-20/7,-40/7],[0,-40/7,-80/7]]`
Dividing `R_2` by `-20/7`
`=[[1,6/7,-2/7],[0,1,2],[0,-40/7,-80/7]]`
`R_1 larr R_1 - 6/7 * R_2`
`=[[1,0,-2],[0,1,2],[0,-40/7,-80/7]]`
`R_3 larr R_3 + 40/7 * R2`
`=[[1,0,-2],[0,1,2],[0,0,0]]`
The system associated with the eigenvalue `lamda=15`
`(A-15I)[[x_1],[x_2],[x_3]]=[[1,0,-2],[0,1,2],[0,0,0]][[x_1],[x_2],[x_3]]=[[0],[0],[0]]`
`i.e.`
`x_1-2x_3=0`
`x_2+2x_3=0`
`i.e.`
`x_1=2x_3`
`x_2=-2x_3`
`:.` eigenvector corresponding to the eigenvalue `lamda=15` is
`v=[[2x_3],[-2x_3],[x_3]]`
Let `x_3=1`
`v_3=[[2],[-2],[1]]`
This material is intended as a summary. Use your textbook for detail explanation.
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