Kunze Linear Algebra · Problem 10.4.12

Bilinear Forms

Mathematics — Methods · preserving-bilinear-form-matrix

Problem

Let FF be a subfield of CC, VV the space of n×1n \times 1 matrices over FF, AA an invertible n×nn \times n matrix over FF, and ff the bilinear form on VV given by f(X,Y)=XtAYf(X, Y) = X^t A Y. If MM is an n×nn \times n matrix over FF, show that MM preserves ff if and only if A−1MtA=M−1A^{-1} M^t A = M^{-1}.

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