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by Ciro Santilli (@cirosantilli, 37)

Matrix representation of a symmetric bilinear form

 ... Algebra Linear algebra Linear map Multilinear map Symmetric bilinear map Symmetric bilinear form
 0 By others on same topic  0 Discussions  Updated 2025-05-26  +Created 1970-01-01  See my version
Like the matrix representation of a bilinear form, it is a matrix, but now the matrix has to be a symmetric matrix.
We can then immediately see that the matrix is symmetric, then so is the form. We have:
B(x,y)=xTMy
(1)
But because B(x,y) is a scalar, we have:
B(x,y)=B(x,y)T
(2)
and:
B(x,y)=B(x,y)T=(xTMy)T=yTMTx=yTMTx=yTMx=B(y,x)
(3)

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  • Dot product
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  • What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms

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