Problem 1: Let F = R or C. For A, B € Mmxn (F), define (A, B) = Tr(AB*), (1) where B* € Mnxm (F) is the conjugate transpose of B, and Tr denotes the trace function, i.e., sum of all diagonal entries of a square matrix. Show that (1) defines an inner product on Mmxn (F).

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
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Problem 1: Let F = R or C. For A, B € Mmxn (F), define
(A, B) = Tr(AB*),
(1)
where B* E Mnxm (F) is the conjugate transpose of B, and Tr denotes the trace
function, i.e., sum of all diagonal entries of a square matrix. Show that (1) defines
an inner product on Mmxn(F).
Transcribed Image Text:Problem 1: Let F = R or C. For A, B € Mmxn (F), define (A, B) = Tr(AB*), (1) where B* E Mnxm (F) is the conjugate transpose of B, and Tr denotes the trace function, i.e., sum of all diagonal entries of a square matrix. Show that (1) defines an inner product on Mmxn(F).
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