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Student[LinearAlgebra]

 BilinearForm
 compute the general bilinear form of two Vectors relative to a Matrix

 Calling Sequence BilinearForm(U, V, A, options)

Parameters

 U, V - Vectors A - (optional) Matrix; defines the bilinear form options - (optional) parameters; for a complete list, see LinearAlgebra[BilinearForm]

Description

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

 Orientation of U Orientation of V Result column column ${U}^{\mathrm{%T}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}V$ row column $U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}V$ row row $U\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{A}^{\mathrm{%T}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{V}^{\mathrm{%T}}$ column row ${U}^{\mathrm{%T}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{A}^{\mathrm{%T}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{V}^{\mathrm{%T}}$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $U≔⟨4|3|2⟩$
 ${U}{≔}\left[\begin{array}{ccc}{4}& {3}& {2}\end{array}\right]$ (1)
 > $V≔⟨1,2,3,4⟩$
 ${V}{≔}\left[\begin{array}{r}{1}\\ {2}\\ {3}\\ {4}\end{array}\right]$ (2)
 > $A≔⟨⟨1,5,w⟩|⟨2,6,x⟩|⟨3,7,y⟩|⟨4,8,z⟩⟩$
 ${A}{≔}\left[\begin{array}{cccc}{1}& {2}& {3}& {4}\\ {5}& {6}& {7}& {8}\\ {w}& {x}& {y}& {z}\end{array}\right]$ (3)
 > $\mathrm{BilinearForm}\left(U,V,A\right)$
 ${330}{+}{2}{}{w}{+}{4}{}{x}{+}{6}{}{y}{+}{8}{}{z}$ (4)