# Thread: Matrix of a bilinear form

1. ## Matrix of a bilinear form

Hi,

I've some troubles finding a matrix of a bilinear form:

Given n >= 1, n in IN and beta : M_{n}(\mathbf{R}) X M_{n}(\mathbf{R}) ----> IR the application defined by beta( A,B)=Tr({t}^A,B) \forall A,B \in M_{n}(\mathbf{R}
(IN= positive integers, IR= real numbers).

Find the matrix of beta in the standar basis of M_{n}(\mathbf{R} .

So I'm stuck here because I don't see what kind of n by n matrix would give me

A*(matrix)*B=\beta (A,B).

I am really sorry the Latex compiler doesn't seem to work .

2. have you tried computing β on basis elements of Mn(R), considered as elements of R^(n^2)? for example, in M2(R), we can identify

[0 0]
[1 0] with e3 in R^4.

you should get an n^2 x n^2 matrix, since dim(Mn(R)) = n^2.

3. Denote B = { E_{ ij } } the standard basis in M_n ( IR ) . Prove that

beta ( E_{ ij} , E_{ ij} ) = 1

beta ( E_{ ij} , E_{ kh} ) = 0 if ( i , j ) =/= ( k , h )

As a consequence, the matrix of beta in the standard basis of M_n ( IR ) is I_{ n^2} .

4. the seaside opening of the river kronecker rises again

5. Originally Posted by Deveno
the seaside opening of the river kronecker rises again