Examples for
Matrix Decompositions
Matrix decompositions are a collection of specific transformations or factorizations of matrices into a specific desired form. Examples of matrix decompositions that Wolfram|Alpha can compute include diagonalization, Jordan, LU, QR, singular value, Cholesky, Hessenberg and Schur decompositions.
Diagonalization
Explore diagonalizations, including unitary and orthogonal diagonalizations, of a square matrix.
Diagonalize a matrix:
Compute an orthogonal diagonalization of a real symmetric matrix:
Calculate a unitary diagonalization of a normal matrix:
LU Decomposition
Decompose a matrix into the product of a lower-triangular matrix and an upper-triangular matrix.
Compute the LU decomposition of a matrix:
Cholesky Decomposition
Decompose a positive definite Hermitian matrix into the product of a lower-triangular matrix and its conjugate transpose.
Find the Cholesky decomposition of a matrix:
Jordan Decomposition
Find the Jordan canonical form of a square matrix.
Compute a Jordan decomposition:
QR Decomposition
Decompose a matrix into the product of a unitary matrix and an upper-triangular matrix.
Compute the QR decomposition of a matrix:
Hessenberg Decomposition
Factor a matrix into the product of a unitary matrix, a Hessenberg matrix with zero entries below the second diagonal and the inverse of the unitary matrix.
Compute a Hessenberg decomposition:
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Singular Value Decomposition
Decompose a matrix into the product of a unitary matrix, a diagonal matrix (of singular values) and another unitary matrix.
Compute a singular value decomposition (SVD):
Schur Decomposition
Decompose a square matrix into the product of an upper-triangular matrix and an orthogonal or unitary matrix.