The operator norm of a linear operator is the largest value by which stretches an element of ,
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It is necessary for and to be normed vector spaces. The operator norm of a composition is controlled by the norms of the operators,
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When is given by a matrix, say , then is the square root of the largest eigenvalue of the symmetric matrix , all of whose eigenvalues are nonnegative. For instance, if
(3)
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then
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which has eigenvalues , so .
The following Wolfram Language code will determine the operator norm of a matrix:
OperatorNorm[a_List?MatrixQ] := Sqrt[Max[Eigenvalues[Transpose[a].a]]]