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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.
SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live [2] or SymPy Gamma. [3] SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies.
An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula d ( A , B ) = ‖ A B → ‖ . {\displaystyle d(A,B)=\|{\overrightarrow {AB}}\|.}
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm.Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
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.
Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the uniform norm on . Note that the definition of uniform norm does not rely on any additional structure on the set X {\displaystyle X} , although in practice X {\displaystyle X} is often at least a ...
OGRePy [34] is Python port of the Mathematica package OGRe (see § Software for use with Mathematica), released in 2024 for Python 3.12 and later. It utilizes SymPy for symbolic computations and Jupyter as a notebook interface. OGRePy allows calculating arbitrary tensor formulas using any combination of addition, multiplication by scalar, trace ...
The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...