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SVD factors a matrix into three matrices U, Σ, and V*, where U and V* are orthogonal and Σ is a diagonal matrix containing the singular values. It's used in signal processing and data compression.

Jordan Decomposition transforms a matrix into its Jordan normal form, which comprises Jordan blocks along the diagonal. It is useful in the theoretical study of linear differential equations.

QR Decomposition factors a matrix into an orthogonal matrix Q and an upper triangular matrix R. This decomposition is used in solving linear least squares problems and for eigenvalue computation.

Cholesky Decomposition represents a Hermitian, positive-definite matrix as the product of a lower triangular matrix and its conjugate transpose. It's used in optimization and monte carlo simulations.

Schur Decomposition factors a matrix into a unitary matrix Q and an upper triangular matrix T. All eigenvalues of the original matrix are located on the diagonal of T. It is used for matrix exponentiation and in control theory.

LU Decomposition factors a matrix as the product of a lower triangular matrix L and an upper triangular matrix U. It's used to solve linear equations, invert matrices, and compute determinants.

Eigendecomposition decomposes a matrix into a product of its eigenvectors and eigenvalues. It's essential in principal component analysis and solving differential equations.

Hessenberg Decomposition converts a matrix into a Hessenberg form which resembles an upper triangular matrix, but has one additional nonzero line below the main diagonal. This form is useful for computing eigenvalues of a matrix.

LDL Decomposition breaks down a Hermitian, positive-definite matrix into LDL*, where L is a lower triangular matrix with unit diagonal and D is a diagonal matrix. It's used for numerical methods in linear systems where pivoting is not desired.

LQ Decomposition is a counterpart to the QR decomposition where a matrix is factored into a lower triangular matrix L and an orthogonal matrix Q. It is used in certain numerical linear algebra applications.

Tridiagonal Decomposition is used predominantly for symmetric matrices, transforming them into a tridiagonal matrix that only has non-zero elements on the main diagonal and the first super and subdiagonals. It simplifies the calculation of eigenvalues.

Matrix Diagonalization is a type of eigendecomposition where a matrix is expressed as the product of its eigenvectors and a diagonal matrix of eigenvalues, assuming it has a full set of distinct eigenvectors. It is widely used for matrix computations like powers and exponentials.

Polar Decomposition factorizes a matrix into a unitary matrix and a positive semi-definite Hermitian matrix. It is used in areas such as quantum mechanics and robotics.

Bidiagonal Decomposition reduces a matrix to a bidiagonal form, consisting of only the main diagonal and either the superdiagonal or the subdiagonal. It is a step in the process of computing SVD.