linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues...

matrix. The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix....

an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can...

of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). The set of all m × n matrices over a field...

Spectral decomposition is any of several things: Spectral decomposition for matrix: eigendecomposition of a matrix Spectral decomposition for linear operator:...

an eigendecomposition of M , {\displaystyle M,} where P {\displaystyle P} is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors...

field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation...

generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any ⁠ m × n {\displaystyle m\times n} ⁠ matrix. It is related...

mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are...

An n×n matrix with n distinct nonzero eigenvalues has 2n square roots. Such a matrix, A, has an eigendecomposition VDV−1 where V is the matrix whose columns...

_{n}} of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is A = U Λ U H {\displaystyle A=U\Lambda...

mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any...

decomposition M = UΣVT, U and V orthogonal, Σ diagonal matrix Eigendecomposition of a symmetric matrix (decomposition according to the spectral theorem) S...

the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix A is said to be...

block-encoding of A with eigendecomposition of a matrix A = ∑ λ i u i u i † {\displaystyle A=\sum \lambda _{i}u_{i}u_{i}^{\dagger }} , one can get a block-encoding...

algorithm. A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of p 1-by-1...

to performing eigendecomposition of a matrix of connectivity weights and assigning systemic importance in proportion to the values of the principal eigenvector...

arbitrary matrices. Eigendecomposition of a matrix Wiener–Khinchin theorem Hawkins, Thomas (1975). "Cauchy and the spectral theory of matrices". Historia...

the covariance matrix. Consider the eigendecomposition of a matrix Σ g = λ g D g A g D g T , {\displaystyle \Sigma _{g}=\lambda _{g}D_{g}A_{g}D_{g}^{T}...

parameter, but it also has the dual role of scale parameter. The eigendecomposition of the matrix M t {\displaystyle M^{t}} yields M i , j t = ∑ l λ l t ψ l...

amount and compute the eigendecomposition of the resulting matrix. In the case when the matrix is depicted as a near-circle, the matrix can be replaced with...

theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur...

algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal...

/ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose...

inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix Q {\displaystyle...

logarithmically which is a heuristic with no guarantees The discrete heat kernel requires eigendecomposition of a matrix of size n × n {\displaystyle...

solution by finding the eigendecomposition of a larger system. For the CARE, we define the Hamiltonian matrix Z = ( A − B R − 1 B ⊤ − Q − A ⊤ ) {\displaystyle...

obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition, which can...

{\displaystyle A} has an eigendecomposition of the form Q E Q T {\displaystyle QEQ^{\mathrm {T} }} where E {\displaystyle E} is a diagonal matrix containing the...

computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based...