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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.
In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
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 .
Another function was called the "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector . [citation needed] Many authors abuse terminology by omitting the quotation marks.
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 ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} , but it is not the same as Schatten norm.
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
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.