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Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix.
Matrix norm – Norm on a vector space of matrices; Norm (mathematics) – Length in a vector space; Normed space – Vector space on which a distance is defined; Operator algebra – Branch of functional analysis; Operator theory – Mathematical field of study
Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ambient matrix spaces. Vladimir Arnold posed [ 16 ] a problem: Find a canonical form of matrices over a field for which the set of representatives of matrix conjugacy classes is a ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
In mathematics, the Smith normal form (sometimes abbreviated SNF [1]) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square ...
The 2-norm of a matrix A is the norm based on the Euclidean vectornorm; that is, the largest value ‖ ‖ when x runs through all vectors with ‖ ‖ =. It is the largest singular value of . In case of a symmetric matrix it is the largest absolute value of its eigenvectors and thus equal to its spectral radius.
The logarithmic norm was independently introduced by Germund Dahlquist [1] and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to nonlinear operators and unbounded operators as well. [2] The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis ...