Eigendecomposition

In linear algebra, eigendecomposition or sometimes spectral decomposition 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.

A (non-zero) vector v of dimension N is an eigenvector of a square (N×N) matrix A if it satisfies the linear equation

where ${\displaystyle \lambda }$ is a scalar, termed the eigenvalue corresponding to v. That is, the eigenvectors are the vectors that the linear transformation A 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

We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an Nth order polynomial equation in the unknown λ. This equation will have N_λ distinct solutions, where 1 ≤ N_λ ≤ N . The set of solutions, that is, the eigenvalues, is called the spectrum of A.

We can factor p as

The integer n_i is termed the algebraic multiplicity of eigenvalue λ_i. The algebraic multiplicities sum to N:

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