How to Find Eigenvalues and Eigenvectors for 2x2 Matrices?

• richard7893
In summary, the conversation is about finding the eigenvalues and eigenvectors for two given matrices. The person is struggling with finding the eigenvectors and is asking for clarification on the process. They also mention that they have found the eigenvalues for one of the matrices, but are unsure about the corresponding eigenvectors. They are wondering how to find the eigenvectors without them being 0.
richard7893

Homework Statement

im trying to find the eigen vector for these 2 matrices: A=[0,0;0,8] AND A=[-8,0;0,0]

The Attempt at a Solution

BACICALLY WHAT IM DOING IS "GUESSING" AT What x1, is then I am coming up wth the solution to x2 once I've made my guess for x1. how can i know for sure if my guyess is correct?

What do you mean by your guesses? Finding eigenvectors and eigenvalues is a process that doesn't need guessing.

Also what do you mean by "the eigenvector for these matrices". Each matrix has two (obvious) eigenvalues and an infinite number of eigenvectors. Do you mean "find the eigenvectors of each matrix" or "find a vector that is an eigenvector for both matrices"?

I mean find the eigenvectors of each matrix. The original question is Find the eigenvals and eigenvecs of A=[1,0;0,9]. I know the eigvals are 1 and 9. However when I try to find the eigvec for lamda =1 and lamda = 9 respectively i get these matrices: when lamda =1 [0,0;0,8] and lamda =9 [-8,0;0,0]. For some reason I'm just thinking that eigenvector for both of these is 0 becase for instance in the mathrix when lamda =1 you get the eqn:
0x+8y=0 and 0x+0y=0. This is almost the same case for when lamda = 9. How do you find x and y to get the eigenvectors without them being 0 because there is no such thing as a 0 eigenvector.

What is a linear algebra eigen vector?

An eigen vector is a non-zero vector that, when multiplied by a square matrix, results in a scalar multiple of itself. It represents the direction in which the matrix has a simple behavior, much like the x and y axes in a graph.

What is the significance of eigen vectors in linear algebra?

Eigen vectors are important in linear algebra because they help us understand the behavior of a matrix. They represent the directions in which the matrix has a simple behavior, and can be used to decompose a matrix into simpler parts for easier analysis.

How do you find eigen vectors?

To find eigen vectors, you first need to find the eigenvalues of a matrix. This can be done by solving the characteristic equation det(A-λI)=0. Once you have the eigenvalues, you can plug them back into the equation (A-λI)x=0 to find the eigen vectors.

Can a matrix have more than one eigen vector?

Yes, a matrix can have multiple eigen vectors. In fact, most matrices have multiple eigen vectors. However, eigen vectors must be linearly independent, meaning they cannot be scaled versions of each other.

What is the relationship between eigen vectors and diagonalization?

Eigen vectors are used in the process of diagonalization, which involves finding a diagonal matrix that is similar to a given matrix. The columns of the diagonal matrix are the eigen vectors of the original matrix. This process is useful for simplifying calculations and solving systems of equations.

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