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Concept
Operator norm
Summary
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . 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 "lengthens" vectors.
Given two normed vector spaces and (over the same base field, either the real numbers or the complex numbers ), a linear map is continuous if and only if there exists a real number such that
The norm on the left is the one in and the norm on the right is the one in .
Intuitively, the continuous operator never increases the length of any vector by more than a factor of Thus the of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators.
In order to "measure the size" of one can take the infimum of the numbers such that the above inequality holds for all
This number represents the maximum scalar factor by which "lengthens" vectors.
In other words, the "size" of is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of as
The infimum is attained as the set of all such is closed, nonempty, and bounded from below.
It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces and .
Every real -by- matrix corresponds to a linear map from to Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all -by- matrices of real numbers; these induced norms form a subset of matrix norms.
If we specifically choose the Euclidean norm on both and then the matrix norm given to a matrix is the square root of the largest eigenvalue of the matrix (where denotes the conjugate transpose of ).
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