Factorization
In mathematics, factorization is the decomposition of an object into several nontrivial factors .
Application examples:
- The always unambiguous prime factorization of a natural number (cf. the factorization procedure to obtain a prime factorization).
- Algebraic terms can often be factored by factoring out and using binomial formulas .
- Polynomials can be factored . Over an algebraically complete field there is always a factorization in linear factors.
- Application for matrices :
- A matrix can be broken down into factors, which is used, for example, when solving systems of linear equations by means of triangular decomposition (also called LU or LR decomposition). In numerical practice, the LR decomposition is usually obtained using the Gaussian elimination method .
- Another matrix factorization from numerics is the QR decomposition , which can normally be obtained using household transformations or Givens rotations .
- In the data analysis , among other things, the non-negative matrix factorization and the binary matrix factorization are considered in order to split matrices into two cluster or concept matrices .
- In a more abstract way, one tries to break down the elements of rings into elementary factors. In addition to number, polynomial and matrix rings, these can also be operator rings.
- In probability theory , factorization is the decomposition of a random variable into independent summands, since the characteristic function of a sum of independent random variables is the product of the individual characteristic functions.
- Spearman's statistical factor analysis .
- The logical factorization of one proposition in relation to another proposition :
- In graph theory , the decomposition of a graph G into subgraphs, in which each node x has only a certain number a of neighboring nodes, is called factorization, and the result is a factor, e.g. B. 1 factors.
Individual evidence
- ^ Karl Popper , David Miller: A proof of the impossibility of inductive probability , in: Nature 302 (1983), 687f.