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The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
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.
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 ...
The Euclidean distance is the prototypical example of the distance in a metric space, [10] and obeys all the defining properties of a metric space: [11] It is symmetric, meaning that for all points and , (,) = (,). That is (unlike road distance with one-way streets) the distance between two points does not depend on which of the two points is ...
The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8 × 10 5.
Both minimize the 2-norm of the residual and do the same calculations in exact arithmetic when the matrix is symmetric. MINRES is a short-recurrence method with a constant memory requirement, whereas GMRES requires storing the whole Krylov space, so its memory requirement is roughly proportional to the number of iterations.
Example. Let X = [X 1, X 2, X 3] be multivariate normal random variables with mean vector ... Calculation of the norm is performed in the L 2 ...
2 Examples. 3 Equivalent definitions. 4 Properties. 5 Table of common operator norms. ... Some common operator norms are easy to calculate, and others are NP-hard.