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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.
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.
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.
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.
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
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 (,) = ‖ ‖.
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 ...
It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let A be a n × n complex matrix. Then the following are equivalent: A is normal. A is diagonalizable by a unitary matrix. There exists a set of eigenvectors of A which forms an orthonormal basis for C n.