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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 ...
Mars Type 14), also known under alternative US designation codename KN-20, [6] is a mobile intercontinental ballistic missile developed by North Korea. It had its maiden flight on 4 July 2017, which coincided with the United States' Independence Day . [ 11 ]
A norm induces a distance, called its (norm) induced metric, by the formula (,) = ‖ ‖. which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space .
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
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.
Start with the definition of a matrix norm, then go through the definitions of induced, Frobenius etc. norms as examples. Then go through the definitions of each property matrix norms might have, with clear results on which norms (do not) possess the given property. In either approach, a table of norms and properties might help the presentation.