There are many different matrix decompositions; each finds use among a particular class of problems.

He was a major contributor to algorithms for matrix decompositions.

He is probably the inventor of the Crout matrix decomposition.

Matrix decomposition techniques have been the most successful.

Some widely used matrix decomposition techniques include the following:

So, if a matrix decomposition of a matrix A is such that:

They are generally referred to as matrix decomposition or matrix factorization techniques.

Matrix decomposition methods simplify computations, both theoretically and practically.

A number of important matrix decompositions involve orthogonal matrices, including especially:

The concept of the polyphase matrix allows matrix decomposition.

Matrix decomposition techniques are data-driven, which avoids many of the drawbacks associated with auxiliary structures.

The Crout matrix decomposition algorithm differs slightly from the Doolittle method.

Matrix decompositions of dense and structured sparse matrices:

Transform techniques (particularly matrix decompositions)

The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.

It does not compute a matrix decomposition, and hence it can be used when A is a very large sparse matrix.

Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix.

GraphLab GraphLab collaborative filtering library, large scale parallel implementation of matrix decomposition methods (in C++) for multicore.

The LUP decomposition algorithm by Cormen et al. generalizes Crout matrix decomposition.

However, classical matrix decompositions like LU and QR decomposition cannot be applied immediately, because the filters form a ring with respect to convolution, not a field.

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

The generalized singular value decomposition (GSVD) is a matrix decomposition more general than the singular value decomposition.

Orthogonal matrix decompositions for dense matrices (QR, RQ, LQ, and QL).

The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.

In machine learning, tree decompositions are also called junction trees, clique trees, or join trees; they play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition.