Let B be a bilinear form on a vector space V over R whose matrix wrt basis (e1,e2) of V is ((1,2),(2,-1)) row-wise. Find a new basis of V wrt which the matrix of B is identity matrix. How to proceed ? Thanks in advance
2026-02-22 19:50:52.1771789852
Bilinear forms by P B Battacharya, Linear Algebra, Chap 7 Example 7.1.10.(4)
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I think that you should seriously doubt the result you're trying to prove (i.e., that such a basis exists). For if $M$ and $N$ are the matrices of a bilinear form $B$ with respect to two different bases, then there's a matrix $Q$ with $$ N = Q^t M Q $$ Since $\det Q^t = \det Q$, that means that the signs of the determinants of $M$ and $N$ are the same. The determinant of the identity matrix is $+1$.