Search results
Results From The WOW.Com Content Network
Scaling (geometry) In affine geometry, uniform scaling (or isotropic scaling [1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed ...
Matrix decomposition. 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.
Determinant. In mathematics, the determinant is a scalar -valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det (A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if ...
Transformation matrix. In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . [citation needed] Note that has rows and columns, whereas the transformation is from to .
Eigenvalues and eigenvectors. In linear algebra, an eigenvector ( / ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: .
Bottom: The action of Σ, a scaling by the singular values σ1 horizontally and σ2 vertically. Right: The action of U, another rotation. In linear algebra, the singular value decomposition ( SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation.
The factorization is not unique: A matrix and its inverse can be used to transform the two factorization matrices by, e.g., [ 53 ] W B B − 1. If the two new matrices W ~ = and H ~ − 1 are non-negative they form another parametrization of the factorization. The non-negativity of W ~ and H ~ applies at least if B is a non-negative monomial ...
Metric multidimensional scaling (mMDS) It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization.