What are the unit eigenvectors for the matrix A = [5 -2; -2 8]?

In summary, the purpose of finding unit eigenvectors is to determine the directions in which a transformation has the greatest effect and to simplify complex calculations. To find unit eigenvectors, one must first find the eigenvalues of the matrix and then solve an equation to obtain corresponding eigenvectors, which are then normalized to have a magnitude of 1. Multiple unit eigenvectors can exist for the same eigenvalue, but they must be normalized to be considered unit eigenvectors. The directions of unit eigenvectors are significant as they represent the principal axes or directions of a transformation and can simplify a complex matrix. In data analysis, unit eigenvectors are commonly used to reduce the dimensionality of a dataset and to interpret and analyze data
  • #1
squaremeplz
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Homework Statement



I'm trying to find the unit eigenvectors corresponding to the following matrix

A = [5 -2; -2 8] ; means new row

Homework Equations



det(A - hI) = 0

The Attempt at a Solution



I get lambda = 4 and 9

unit eigenvector corresponding to lambda = 4

x1 = ( -2/sqrt(5), -1/sqrt(5) )^T

unit eigenvector corresponding to lambda = 9

x2 = (1/sqrt(5), -2/sqrt(5) )^T

Could someone please let me know if they get the same result. Thanks!
 
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FAQ: What are the unit eigenvectors for the matrix A = [5 -2; -2 8]?

What is the purpose of finding unit eigenvectors?

The purpose of finding unit eigenvectors is to determine the directions in which a transformation has the greatest effect. Unit eigenvectors also help in understanding the behavior of a system and can be used to simplify complex calculations.

How do you find unit eigenvectors?

To find unit eigenvectors, you first need to find the eigenvalues of the matrix. Then, for each eigenvalue, you can find the corresponding eigenvector by solving the equation (A-λI)x=0, where A is the original matrix and λ is the eigenvalue. Finally, you can normalize the resulting eigenvectors to have a magnitude of 1.

Can there be multiple unit eigenvectors for the same eigenvalue?

Yes, there can be multiple unit eigenvectors for the same eigenvalue. In fact, for a given eigenvalue, there can be an infinite number of eigenvectors that are all scalar multiples of each other. Therefore, when finding unit eigenvectors, it is important to ensure that the eigenvectors are normalized to have a magnitude of 1.

What is the significance of the unit eigenvectors' directions?

The unit eigenvectors' directions represent the directions in which the transformation has the greatest effect. These directions are also known as the principal axes or principal directions. They are important in understanding the behavior of a system and can be used to transform a complicated matrix into a simpler diagonal matrix.

How are unit eigenvectors used in data analysis?

Unit eigenvectors are commonly used in data analysis to reduce the dimensionality of a dataset. By finding the unit eigenvectors of a covariance matrix, we can identify the most important directions in the data and project the data onto these directions. This helps in visualizing and interpreting the data, as well as in reducing the computational complexity of analyzing the data.

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