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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.
Operator norm. 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. Informally, the operator norm of a linear map is the maximum factor by which it ...
The definition of the condition number depends on the choice of norm, as can be illustrated by two examples. If ‖ ‖ is the matrix norm induced by the (vector) Euclidean norm (sometimes known as the L 2 norm and typically denoted as ‖ ‖), then
Norm (mathematics) 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. In particular, the Euclidean distance in a Euclidean space ...
The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space C ( [ a , b ] ) {\displaystyle C([a,b])} of continuous complex valued functions f {\displaystyle f} and g ...
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable ...
The logarithmic norm is related to the extreme values of the Rayleigh quotient. It holds that. and both extreme values are taken for some vectors . This also means that every eigenvalue of satisfies. . More generally, the logarithmic norm is related to the numerical range of a matrix. A matrix with is positive definite, and one with is negative ...