Eigenvalues and eigenvectors

Eigenvectors are unchanged in direction by a linear transformation

Eigenvectors remain unchanged in direction when a linear transformation is applied. This unique property allows them to be scaled by a constant factor, known as the eigenvalue. Understanding eigenvectors and eigenvalues is crucial in various applications, including solving systems of linear equations and analyzing stability in dynamical systems.

Example

Consider a linear transformation represented by the matrix A = [[2, 0], [0, 3]]. The eigenvector v = [1, 0] remains unchanged in direction when transformed by A, as A * v = [2, 0] = 2 * v. The corresponding eigenvalue λ = 2 indicates that v is scaled by a factor of 2.

Remember this

Recognizing eigenvectors and eigenvalues helps in simplifying complex problems in linear algebra and understanding the behavior of systems over time.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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