Eigendecomposition of a matrix
ID: eigendecomposition-of-a-matrix
Every invertible matrix can be written as:where:Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.
- is a diagonal matrix containing the eigenvalues of
- columns of are eigenvectors of
Intuitively, Note that this is just the change of basis formula, and so:
- changes basis to align to the eigenvectors
- multiplies eigenvectors simply by eigenvalues
- changes back to the original basis
Eigendecomposition is a fundamental concept in linear algebra that involves decomposing a square matrix into its eigenvalues and eigenvectors. Specifically, for a square matrix \( A \), the eigendecomposition is expressed in the following form: \[ A = V \Lambda V^{-1} \] where: - \( A \) is the original \( n \times n \) matrix. - \( V \) is a matrix whose columns are the eigenvectors of \( A \).
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