ProjectionMatrix - Maple Help

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LinearAlgebra

  

ProjectionMatrix

  

construct the matrix of the orthogonal projection onto a subspace

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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=MM*M−1M*

• 

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:

S1,2,3,4,4,3,2,1

S1234,4321

(1)

PProjectionMatrixS

P7102511015253101511011015310251511025710

(2)

v1,0,1,3

v10−13

(3)

wP·v

w012132

(4)

BasisopS,w=S

1234,4321=1234,4321

(5)

w·vw

0

(6)

PProjectionMatrix1,2,3,4,5,6,datatype=float8,shape=symmetric

P0.8333333333333330.333333333333333−0.1666666666666670.3333333333333330.3333333333333330.333333333333333−0.1666666666666670.3333333333333330.833333333333333

(7)

P·1,0,0

0.8333333333333330.333333333333333−0.166666666666667

(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)

Compatibility

• 

The LinearAlgebra[ProjectionMatrix] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

LinearAlgebra

Matrix

Vector