Finding eigenvalues and eigenvectors of a matrix A

In summary, using the formula for finding eigenvalues and eigenvectors, the matrix A = [2, 1; 8, 4] has two eigenvalues of 0 and 6, with corresponding eigenvectors [1; -2] and [1; 4]. The determinant of the matrix is 0, indicating that 0 is an eigenvalue.
  • #1
Rubik
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0

Homework Statement



Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4]

Homework Equations



det(A - lambda I) = 0

The Attempt at a Solution



After expanding using the formula I have the equation (2 - [itex]\lambda[/itex])(4 - [itex]\lambda[/itex]) - 8

Which gives [itex]\lambda[/itex] = 0, 6 (Should I be worried that I got a zero in my answer?)

Then substituting the values for lambda back into the problem and using (A - [itex]\lambda[/itex]I)v = 0

For [itex]\lambda[/itex] = 6
[-4, 1; 8, -2] * [a; b] = [0; 0] and I get the eigenvector a[1; 4] from (-4a + b = 0)

For [itex]\lambda[/itex] = 0
[2, 1; 8, 4] * [a; b] = [0; 0] I get a second eigenvector of a[1; -2] from (2a + b = 0)

Did I get the right eigenvalues and corresponding eigenvectors?
 
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  • #2
Yes, you did. And it's easy for you to check as well. A*[1;4] should give you 6*[1;4] and A*[1;-2] should give you 0*[1;-2], right? Does it work?
 
  • #3
Rubik said:

Homework Statement



Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4]

Homework Equations



det(A - lambda I) = 0

The Attempt at a Solution



After expanding using the formula I have the equation (2 - [itex]\lambda[/itex])(4 - [itex]\lambda[/itex]) - 8

Which gives [itex]\lambda[/itex] = 0, 6 (Should I be worried that I got a zero in my answer?)
Notice that the determinant of the matrix is 2(4)- 1(8)= 0. Since the determinant of a matrix is the product of its eigenvalues, 0 must be an eigenvalue.

Then substituting the values for lambda back into the problem and using (A - [itex]\lambda[/itex]I)v = 0

For [itex]\lambda[/itex] = 6
[-4, 1; 8, -2] * [a; b] = [0; 0] and I get the eigenvector a[1; 4] from (-4a + b = 0)

For [itex]\lambda[/itex] = 0
[2, 1; 8, 4] * [a; b] = [0; 0] I get a second eigenvector of a[1; -2] from (2a + b = 0)

Did I get the right eigenvalues and corresponding eigenvectors?
 

1) What are eigenvalues and eigenvectors in the context of matrices?

Eigenvalues and eigenvectors are a set of numbers and corresponding vectors, respectively, that describe the behavior of a matrix when multiplied by those vectors. Eigenvalues represent the scaling factor of the eigenvectors when they are multiplied by the matrix.

2) How are eigenvalues and eigenvectors calculated for a matrix?

To find the eigenvalues and eigenvectors of a matrix, we solve the characteristic equation (det(A-λI)=0) where A is the matrix, λ is the eigenvalue, and I is the identity matrix. The eigenvalues are the solutions to this equation, and the corresponding eigenvectors can be found by plugging in each eigenvalue into the equation (A-λI)x=0 and solving for x.

3) What is the significance of eigenvalues and eigenvectors in linear algebra?

Eigenvalues and eigenvectors play a crucial role in many applications of linear algebra, such as in solving systems of linear equations, understanding transformations and geometric concepts, and in data analysis and machine learning. They also have connections to other important concepts, such as determinants and diagonalization.

4) Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. This is especially common in matrices with complex entries or in applications involving oscillatory systems. In these cases, the eigenvalues and eigenvectors are complex numbers, but the same principles for finding them still apply.

5) How can the eigenvalues and eigenvectors of a matrix be used to simplify calculations?

By finding the eigenvalues and eigenvectors of a matrix, we can often transform it into a diagonal or triangular form, which can make calculations and manipulations much easier. This is known as diagonalization, and it is a powerful tool in linear algebra and various other fields.

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