construct the matrix of the orthogonal projection onto a subspace - Maple Help

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LinearAlgebra[ProjectionMatrix] - construct the matrix of the orthogonal projection onto a subspace

Calling Sequence

ProjectionMatrix(S, conj, options)

Parameters

S

-

{set, list}(Vector); Vectors spanning the subspace to project onto

conj

-

BooleanOpt(conjugate); (optional) specifies if the Hermitian transpose is used (default: true)

options

-

(optional); constructor options for the result object

Description

• 

The ProjectionMatrix(S) command constructs the matrix of the orthogonal linear projection onto the subspace spanned by the vectors in S.  If B is a maximal, linearly independent subset of S and M is the Matrix whose columns are the Vectors in B, then

ProjectionMatrixS=M.M%H.M-1.M%H

• 

If the conj option is omitted or provided in either of the forms conjugate or conjugate=true, the projection matrix is constructed using Hermitian transpose operations.  If the conj option is given as conjugate=false, the ordinary transpose is used.

• 

Additional arguments are passed as options to the Matrix constructor which builds the result.

Examples

withLinearAlgebra:

S:=1,2,3,4,4,3,2,1

S:=1234,4321

(1)

P:=ProjectionMatrixS

P:=7102511015253101511011015310251511025710

(2)

v:=1,0,1,3

v:=1013

(3)

w:=P.v

w:=012132

(4)

BasisopS,w=S

1234,4321=1234,4321

(5)

w.vw

0

(6)

P:=ProjectionMatrix1,2,3,4,5,6,datatype=float8,shape=symmetric

P:=0.8333333333333330.3333333333333330.1666666666666670.3333333333333330.3333333333333330.3333333333333330.1666666666666670.3333333333333330.833333333333333

(7)

P.1,0,0

0.8333333333333330.3333333333333330.166666666666667

(8)

ProjectionMatrixa,1

aa&conjugate0;a&conjugate0;a+1aa&conjugate0;a+1a&conjugate0;a&conjugate0;a+11a&conjugate0;a+1

(9)

ProjectionMatrixa,1,conjugate=false

a2a2+1aa2+1aa2+11a2+1

(10)

See Also

LinearAlgebra, Matrix, Vector


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