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Student[LinearAlgebra]

  

BilinearForm

  

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

 

Calling Sequence

Parameters

Description

Examples

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, UTypesetting:-_Hold%T, and V' is either V or its transpose, VTypesetting:-_Hold%T, according to the following rules:

Orientation of U

Orientation of V

  Result

     column

     column

UTypesetting:-_Hold%T·A·V

      row

     column

U·A·V

      row

      row

U·ATypesetting:-_Hold%T·VTypesetting:-_Hold%T

     column

      row

UTypesetting:-_Hold%T·ATypesetting:-_Hold%T·VTypesetting:-_Hold%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

withStudentLinearAlgebra:

U4|3|2

U432

(1)

V1,2,3,4

V1234

(2)

A1,5,w|2,6,x|3,7,y|4,8,z

A12345678wxyz

(3)

BilinearFormU,V,A

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

(4)

See Also

LinearAlgebra[BilinearForm]

Student[LinearAlgebra]

Student[LinearAlgebra][Operators]

Student[LinearAlgebra][SetDefault]