Operator Norm

The operator norm of a linear operator T:V->W is the largest value by which T stretches an element of V,


It is necessary for V and W to be normed vector spaces. The operator norm of a composition is controlled by the norms of the operators,


When T is given by a matrix, say T(v)=Av, then ||T|| is the square root of the largest eigenvalue of the symmetric matrix A^(T)A, all of whose eigenvalues are nonnegative. For instance, if

 A=[2 0 0; 3 0 2]


 A^(T)A=[13 0 6; 0 0 0; 6 0 4],

which has eigenvalues {0,1,16}, so ||A||=4.

The following Wolfram Language code will determine the operator norm of a matrix:

  OperatorNorm[a_List?MatrixQ] :=

This entry contributed by Todd Rowland

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Rowland, Todd. "Operator Norm." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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