Spectral Decomposition
Spectral decomposition is any of several things: * Spectral decomposition for matrix: eigendecomposition of a matrix * Spectral decomposition for linear operator: spectral theorem *Decomposition of spectrum (functional analysis) The spectrum of a linear operator T that operates on a Banach space X is a fundamental concept of functional analysis. The spectrum consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectru ...

Eigendecomposition Of A Matrix
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 A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form \mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues p\left(\lambda\right) = ...
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