Find the Eigenvalues of the matrix and a corresponding eigenvalue

In summary, we found the eigenvalues of the given matrix to be 5 and 0, with corresponding eigenvectors (-2, 1) and (1/2, 1). To check if these eigenvectors form an orthogonal set of vectors, we can look at the definition of orthogonal and see that they do indeed satisfy the conditions. We also discussed the definition of diagonalizable matrix and the importance of finding an orthogonal matrix to diagonalize the given matrix. Lastly, we determined that the given eigenvectors are already orthogonal and can be normalized to form the columns of an orthogonal matrix.
  • #1
suspenc3
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0
Find the Eigenvalues of the matrix and a corresponding eigenvalue. Check that the eigenvectors associated with the distinct eigenvalues are orthogonal. Find an orthogonal matrix that diagonalizes the matrix.

(1)[tex]\left(\begin{array}{cc}4&-2\\-2&1\end{array}\right)[/tex]

I found my eigenvalues to be 5 & 0, and the corresponding eigenvectors to be

[tex]\left(\begin{array}{cc}-2\\1\end{array}\right)[/tex]

and

[tex]\left(\begin{array}{cc}1/2\\1\end{array}\right)[/tex]

The book doesn't really explain this section well, can someone help me out with what to do next?

Also, how do you know if the eigenvectors produce an orthogonal set of vectors?what is orthonormal?

Thanks
 
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  • #2
you are correct in the eigenvectorsWhat is the definition of orthogonal? An [itex] n\times n [/itex] invertible matrix [tex] Q [/tex] is called orthogonal, iff [tex] Q^{-1} = Q^{T} [/tex]

In other words, [tex] QQ^{T} = I [/tex]

An orthonormal set of vectors is basically a set of mutually orthogonal vectors such that: (1) [tex] u_{i}\cdot u_{i} = 1 [/tex] and (2) [tex] u_{i}\cdot u_{j} = 0 [/tex]

To see if the eigenvectors produce an orthogonal set of vectors, look at the definiton of orthogonal again.

What is a diagonalizable matrix?
 
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  • #3
A matrix is diagonalizable if there exists an n X n non-singular matrix P such that [tex]P^-^1AP[/tex] is a diagonal matrix.

If I take [tex]AA^t[/tex] I should get the Identity shouldn't I?I don't think it works..but it is a symmetric matrix..so does that mean that there is always an orthogonal matrix that diagonalizes A?
 
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  • #4
No, you shouldn't. Why would you think that? As courtigrad said, that's the definition of "orthogonal matrix" and you are not told that A is orthogonal. Yes, a matrix A is diagonalizable if there exist a matrix P such that PAP-1 is diagonal. You are asked to find an orthogonal matrix that "diagonalizes" A. It is P you want to be orthogonal, not A.

The eigenvectors you got, (-2, 1) and (1/2, 1) are already orthogonal to one another: [itex](-2, 1)\cdot(1/2,1)= (-2)(1/2)+ 1(1)= 0][/itex]. Now can you find vectors of length 1 in the same direction? Those will give you the columns of an orthognal matrix.
 
  • #5
Ohh, alright

Thanks
 

Related to Find the Eigenvalues of the matrix and a corresponding eigenvalue

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts that are used to analyze linear transformations. Eigenvalues are numbers that represent the scaling factor of the eigenvectors after the linear transformation is applied.

2. Why is it important to find the eigenvalues of a matrix?

Finding the eigenvalues of a matrix is important because it allows us to understand the behavior of the matrix and its associated linear transformation. It also helps in solving systems of linear equations and in diagonalizing matrices.

3. How do you find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, we need to solve the characteristic equation det(A-λI)=0, where A is the given matrix and λ is the eigenvalue. This equation will give us the eigenvalues of the matrix.

4. Can a matrix have more than one eigenvalue?

Yes, a matrix can have multiple eigenvalues. The number of unique eigenvalues of a matrix is limited by the size of the matrix, but there can be repeated eigenvalues.

5. What is the significance of the eigenvalues of a matrix?

The eigenvalues of a matrix help us understand the properties and behavior of the matrix. They can help in solving systems of linear equations, finding the inverse of a matrix, and in understanding the structure of a matrix. They also have applications in various fields such as physics, engineering, and computer science.

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