Concept

# Eigendecomposition of a matrix

Summary
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form :\mathbf{A} \mathbf{v} = \lambda \mathbf{v} for some scalar λ. Then λ is called the eigenvalue corresponding to v. Geometrically speaking, the eigenvectors of A are the vectors that A merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigen
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