Why Can't I Find All Eigenvectors for My 3x3 Matrix?

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SUMMARY

The discussion centers on finding all eigenvectors for the 3x3 matrix A = {{7,-5,0},{-5,7,0},{0,0,-6}}. The eigenvalues identified are 2, 12, and -6, but the user only derived one eigenvector, (0,0,1). The user acknowledges the existence of two additional eigenvectors, (-1,1,0) and (1,1,0), but struggles to compute them manually. A recommended resource for further understanding is the Khan Academy tutorial on eigenvectors and eigenspaces for 3x3 matrices.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix operations, specifically (A-λI)
  • Basic knowledge of linear algebra concepts
  • Experience with solving homogeneous systems of equations
NEXT STEPS
  • Study the process of calculating eigenvectors using the characteristic polynomial
  • Learn about eigenspaces and their geometric interpretations
  • Explore the implications of repeated eigenvalues on eigenvector multiplicity
  • Review the tutorial "Eigenvectors and eigenspaces for a 3x3 matrix" on Khan Academy
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as anyone involved in computational mathematics or engineering applications requiring eigenvalue analysis.

haz
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Hey everyone.
I have a matrix A = {{7,-5,0},{-5,7,0},{0,0,-6}}
I have found the Eigenvalues, 2,12,-6 but I'm only getting one Eigenvector, (0,0,1)..
I know there is 2 others (-1,1,0) and (1,1,0) but I am unable to get them by hand.
Once I get the matrix in the form (A-λI)*v = o, I just get the one vector..
Any help is much appreciated.
 
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