What is the Correct Way to Find Eigenvalues and Eigenvectors of a Matrix?

Click For Summary
SUMMARY

The correct method to find the eigenvalues and eigenvectors of the matrix A = [-4 4 8; 0 0 -10; 0 0 2] involves calculating the characteristic polynomial by setting A - λI = 0. The eigenvalues determined from this calculation are λ = 0 (with multiplicity 2) and λ = 1. The corresponding eigenvectors are (1, 1, 0) for λ = 0 and (1, 0, 0) for λ = 1. It is crucial not to reduce the matrix before finding the eigenvalues, as this alters the results except for the zero eigenvalue.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix operations
  • Knowledge of characteristic polynomials
  • Proficiency in linear algebra concepts
NEXT STEPS
  • Study the derivation of characteristic polynomials for matrices
  • Learn about the implications of matrix reduction on eigenvalues
  • Explore methods for calculating eigenvectors from eigenvalues
  • Investigate applications of eigenvalues and eigenvectors in various fields
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as engineers and data scientists who utilize eigenvalues and eigenvectors in their work.

Seon
Messages
1
Reaction score
0
Find the eigenvalues and corresponding eigenvector of the matrix.
A=
[-4 4 8 ]
[0 0 -10]
[0 0 2 ]

[1 -1 0]
~ [0 0 1 ]
[0 0 0 ]

I calculated by A = -\lambdaI

So,

[1-lamda -1 0 ]
[0 -lamda 1]
[0 0 -lamda]

so, lamda = 0,0, and 1

So I got

1st eigen value: 0 eigen vector (1,1,0)
2nd eigen value: 0 eigen vector (1,1,0)
3rd eigen value: 1 eigen vector (1,0,0)

1st and 2nd values were right, but third one was wrong.
I tried several times, and I always get 1(1,0,0)

What do i need to do ?
thanks
 
Physics news on Phys.org
if you reduce the matrix, you change the eigenvalues, except for 0. don't reduce the matrix, find the characteristic polynomial of the original A.
 

Similar threads

Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Find the Eigenvalues and eigenvectors of 3x3 matrix
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad The Orthogonality of the Eigenvectors of a 2x2 Hermitian Matrix
  • · Replies 13 ·
Replies
13
Views
4K
Graduate Eigenvector of Pauli Matrix (z-component of Pauli matrix)
  • · Replies 2 ·
Replies
2
Views
3K
MHB Solving Matrix A: Characteristic Equation and Eigenvectors
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Is there always a matrix corresponding to eigenvectors?
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Can anyone explain to me why eigenvector here is like this
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Eigenvalues and Eigenvectors question
  • · Replies 16 ·
Replies
16
Views
2K