Finding a normalized eigenvector

in my problem i have 2 eigenvectors, (1,3) and (3,1)

(1,3) corresponds to eigenvalue 10

(3,1) corresponds to eigenvalue 20

in my notes i have written 'to normalize make equal to one and solve', of course this was about a month ago and i have completely forgotten what i meant by that.

any help appreciated!

A normalized eigenvector is a vector that has been scaled to have a length or magnitude of 1. It is a special type of eigenvector that is useful in certain mathematical calculations.

In many scientific and mathematical applications, normalized eigenvectors are used to represent important characteristics of a system or dataset. They can also simplify complex calculations and make them more manageable.

To find a normalized eigenvector, you first need to find the corresponding eigenvalue of a given matrix. Then, you can use the eigenvalue and the matrix to solve for the eigenvector. Finally, you can normalize the eigenvector by dividing each element by the magnitude of the vector.

An eigenvector is any vector that, when multiplied by a matrix, results in a scalar multiple of itself. A normalized eigenvector is a specific type of eigenvector that has been scaled to have a magnitude of 1.

Yes, a matrix can have multiple normalized eigenvectors associated with different eigenvalues. In fact, a matrix can have an infinite number of normalized eigenvectors if it is a diagonalizable matrix.