Eigenvalues and eigenvectors - Eigenvalue and Eigenvectors

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Eigenvalues and eigenvectors

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Notes:
An eigenvector of an n×nn \times n matrix AA is a non-zero vector xx such that Ax=λxAx= \lambda x, for some scalar λ\lambda. The scalar λ\lambda is called the eigenvalue.

We say the eigenvector xx corresponds to the eigenvalue λ\lambda.

Given an eigenvalue λ\lambda of matrix AA, we can find a corresponding eigenvector xx by solving
(AλI)x=0(A-\lambda I)x=0
And finding a non-trivial solution xx.

The eigenspace is the null space of the matrix AλIA-\lambda I. In other words, the eigenspace is a set of all solutions for the equation
(AλI)x=0(A-\lambda I)x=0

Of course, we can find the basis for the eigenspace by finding the basis of the null space of AλIA-\lambda I.
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Eigenvalues and eigenvectors

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