The BilinearForm(U, V, A) command computes the product $U\'\.A\.V\'$, where $U\'$ is either U or its transpose, $U^{\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%T}\right]\right)}$, and $V\'$ is either V or its transpose, $V^{\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%T}\right]\right)}$, according to the following rules:

Note: The orientation of V solely determines whether the Matrix A is transposed.

If the conjugate option is specified, or globally set through the SetDefault command, the rules are slightly different.  See LinearAlgebra[BilinearForm] for details.

If A is omitted, then it defaults to the identity Matrix, and the bilinear form is the dot product.

The dimensions of U, V, and A must be such that the product can be formed.  In particular, if A is not included in the calling sequence for bilinear form, U and V must have the same dimension.

By default in the Student[LinearAlgebra] package, complex conjugates are not used when forming dot products, including when computing bilinear forms.  This behavior can be modified with the SetDefault command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$U≔⟨4\left|3\right|2⟩$

$U≔\left[\begin{array}{ccc}4& 3& 2\end{array}\right]$

$V≔⟨1\,2\,3\,4⟩$

$V≔\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$

$A≔⟨⟨1\,5\,w⟩\left|⟨2\,6\,x⟩\right|⟨3\,7\,y⟩|⟨4\,8\,z⟩⟩$

$A≔\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& 7& 8\\ w& x& y& z\end{array}\right]$

$\mathrm{BilinearForm}\left(U\,V\,A\right)$

$330+2w+4x+6y+8z$