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A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1]
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}}\|.}
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.
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.
Dual norm – Measurement on a normed vector space; 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
The first goal is to find invertible square matrices and such that the product is diagonal. This is the hardest part of the algorithm. Once diagonality is achieved, it becomes relatively easy to put the matrix into Smith normal form.
If is a reflexive Banach space then this conclusion is also true when = [2]. Metric reformulation. As usual, let (,):= ‖ ‖ denote the canonical metric induced by the norm, call the set {: ‖ ‖ =} of all vectors that are a distance of from the origin the unit sphere, and denote the distance from a point to the set by (,) := (,) = ‖ ‖.
The perimeter of the square is the set of points in ℝ 2 where the sup norm equals a fixed positive constant. For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.