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In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. [7] The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject. [8]
This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. [6] The algorithm was further optimised in 2017, [7] reducing the number of matrix additions per step to 12 while maintaining the number of matrix multiplications, and again in ...
Given an input matrix and a desired low rank , the randomized LU returns permutation matrices , and lower/upper trapezoidal matrices , of size and respectively, such that with high probability ‖ ‖ +, where is a constant that depends on the parameters of the algorithm and + is the (+)-th singular value of the input matrix . [14]
Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach. The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations).
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.