16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.1.6

Bilinear Forms

Mathematics — Linear Algebra · bilinear-form-trace

Problem

Let nn be a positive integer, and let VV be the space of all n×nn \times n matrices over the field of complex numbers. Show that the equation f(A,B)=n tr (AB)−tr (A) tr (B)f(A, B) = n\,\text{tr}\,(AB) - \text{tr}\,(A)\,\text{tr}\,(B) defines a bilinear form ff on VV. Is it true that f(A,B)=f(B,A)f(A, B) = f(B, A) for all A,BA, B?

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