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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.
Where is the intersection (i.e. the dot product) of the document (d 2 in the figure to the right) and the query (q in the figure) vectors, ‖ ‖ is the norm of vector d 2, and ‖ ‖ is the norm of vector q.
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 (,) = ‖ ‖.
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.
Matrix norm, a map that assigns a length or size to a matrix; Operator norm, a map that assigns a length or size to any operator in a function space; Norm (abelian group), a map that assigns a length or size to any element of an abelian group; Field norm a map in algebraic number theory and Galois theory that generalizes the usual distance norm
This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis.
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