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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.
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...
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 ...
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 ...
Every real -by-matrix corresponds to a linear map from to . Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all -by-matrices of real numbers; these induced norms form a subset of matrix norms.
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.
Notice that ‖ ‖ is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), ‖ ‖ is the trace class norm (see trace class), and ‖ ‖ is the operator norm (see operator norm). Note that the matrix p-norm is often also written as ‖ ‖, but it is not the same as Schatten norm.
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 ...