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

In summary, eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of a square matrix. Eigenvalues are scalar values that represent the scaling factor of the eigenvector when the matrix is applied to it. Eigenvectors are non-zero vectors that remain in the same direction after being multiplied by the matrix. They are important because they provide insights into the behavior of a matrix and are used in various applications such as solving linear equations and dimensionality reduction in data analysis. The calculation of eigenvalues and eigenvectors involves finding the values that satisfy the characteristic equation for a given matrix. A matrix can have multiple eigenvalues and eigenvectors, with the number of eigenvalues and eigenvectors being equal to its
  • #1
Seon
1
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 = -[itex]\lambda[/itex]I

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
 
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  • #2
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.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of a square matrix. Eigenvalues are scalar values that represent the scaling factor of the eigenvector when the matrix is applied to it. Eigenvectors are non-zero vectors that remain in the same direction after being multiplied by the matrix.

2. Why are eigenvalues and eigenvectors important?

Eigenvalues and eigenvectors are important because they provide insights into the behavior of a matrix. They are used to solve systems of linear equations, determine stability of dynamic systems, and perform dimensionality reduction in data analysis.

3. How are eigenvalues and eigenvectors calculated?

The calculation of eigenvalues and eigenvectors involves finding the values that satisfy the characteristic equation for a given matrix. This can be done using various methods such as the power method, QR algorithm, or diagonalization.

4. Can a matrix have multiple eigenvalues and eigenvectors?

Yes, a matrix can have multiple eigenvalues and eigenvectors. In fact, a square matrix will have the same number of eigenvalues and eigenvectors as its dimensions. However, repeated eigenvalues may correspond to multiple eigenvectors.

5. How are eigenvalues and eigenvectors used in data analysis?

In data analysis, eigenvalues and eigenvectors are used in techniques such as principal component analysis (PCA) to reduce the dimensionality of a dataset. This allows for easier visualization and analysis of the data. Eigenvalues and eigenvectors can also be used to identify patterns and relationships in data.

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