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Eigenvectors
Eigenvectors An eigenvector of a linear transformation is a nonzero vector whose image is unchanged by the linear transformation. Eigenvectors can be though...
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Eigenvectors An eigenvector of a linear transformation is a nonzero vector whose image is unchanged by the linear transformation. Eigenvectors can be though...
Eigenvectors
An eigenvector of a linear transformation is a nonzero vector whose image is unchanged by the linear transformation. Eigenvectors can be thought of as vectors that are transformed into themselves by the linear transformation. The corresponding eigenvalue is a scalar value associated with the eigenvector, representing the scale at which the vector is stretched or compressed.
Key Properties of Eigenvectors and Eigenvalues:
An eigenvector corresponding to a given eigenvalue is unique.
The sum of the eigenvalues of a linear transformation is equal to the dimension of the domain.
Eigenvectors are often used to solve linear systems of equations and to understand the behavior of linear transformations.
Examples:
Consider the linear transformation T: R² → R², where T(x, y) = (x + y, x - y).
The eigenvector (1, 1) of T corresponds to the eigenvalue 1. This is because T(1, 1) = (2, 0), which is unchanged by T.
The eigenvector (-1, 1) of T corresponds to the eigenvalue -1. This is because T(-1, 1) = (-1, 0), which is also unchanged by T.
Applications of Eigenvectors:
Eigenvectors and eigenvalues are used in various applications in linear algebra, including:
Determining the eigenvalues and eigenvectors of a matrix.
Solving linear systems of equations.
Understanding the behavior of linear transformations.
Solving optimization problems