Objectives

• The main objective of matrix factorization is to reduce matrix systems into basic building blocks or factors, such that they can be better understood and used for other applications. Among the main examples on which matrix factorization methods are used are orthogonal or unitary matrix and a triangular matrix.
  
  

Significance

• Matrix factorization is important for various pure and applied mathematical issues. It is used in algebra, geometry, physics and astronomy. In theoretical physics, particles in the universe are regarded to be part of a world-volume of branes, and interaction between different kinds of branes are measured by means of matrix factorization.
  
  

Examples

• Examples of matrix factorization can be found in linear equations, which can be further defined by means of LU decomposition. A system of the linear equation Ax = b, A is a matrix that can be decomposed through factoring, in turn coming up with an upper triangular and lower triangular matrix. Thus, the systems Ux = L = 1b and L(Ux) = b become the factor matrices of A and require less addition and multiplication operations to be solved. Likewise, the QR factorization expresses A as a product of QR, with Q expressed as a unitary matrix and R an upper triangular matrix. Applying the previous system, Ax = b can be expressed as Q(Rx) = b and is solved by Rx = QTb = e, while Rx = e can be solved through back substitution.
  
  

Variations

• Matrix factorization is expressed in various forms, depending on the operation and objective of the study. These include the LU reduction, which is used for super computing and parallel computing, while the Cholesky decomposition is used to factor out the square root of a given matrix. Block LU decomposition is applied in block mathematics, while single-value decomposition is applied when factoring real or complex matrices to determine values such as rank, range and null space.
  
  

Applications

• Aside from theoretical physics, the matrix factorization method is also important for more practical purposes, such as statistics and signal processing. Defining symmetry among matrices is also required in quantum theory, pattern recognition and modal analysis.