Advanced Linear Algebra, Lecture 5.7: The norm of a linear map

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Advanced Linear Algebra, Lecture 5.7: The norm of a linear map

The set Hom(X,U) of linear maps from X to U is itself a vector space, and so we can ask how to put an inner product structure or a norm on it. There are numerous ways to do this, and we introduce two of them. The Frobenius norm arises from the inner product (A,B)=tr(B*A) and is independent of the inner product structure on X or U. The induced norm, defined by ||A|| = max ||Ax||/||x||, depends on both inner product structures. In the remainder of the lecture, we prove some basic properties about this norm, and show that ||A||=||A*||. We also show that the invertible maps form an open subset of Hom(X,U). We conclude with a general definition of a norm of a linear map.