Overview of Computation in Student[LinearAlgebra] - Maple Help

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Overview of Computation in Student[LinearAlgebra]

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For a general introduction to the LinearAlgebra subpackage of the Student package and a list of the linear algebra computation routines, see Student[LinearAlgebra].

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The computation routines in the Student[LinearAlgebra] subpackage are interfaces to the corresponding routines in the top-level LinearAlgebra package.  There are two principal differences that these interfaces implement, however.

  

First, the top-level LinearAlgebra routines use hardware floating-point computations whenever possible.  While this is important for large scale problems, it is potentially confusing, so in the Student[LinearAlgebra] subpackage this feature is turned off by default.

  

Second, the top-level LinearAlgebra routines generally treat symbols as complex-valued rather than real-valued.  For example, a calculation such as a,b.c,d results in complex conjugates being applied to some of the symbols.  Again, this working environment, while important in the context of the full Maple program, is less essential in the Student[LinearAlgebra] context, and symbols are generally treated as real-valued in this package.

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The commands available for use in linear algebra computation are:

&x

*

+

.

^

^*

^+

AddRow

Adjoint

BackwardSubstitute

BandMatrix

Basis

BilinearForm

CharacteristicMatrix

CharacteristicPolynomial

ColumnDimension

ColumnSpace

CompanionMatrix

ConstantMatrix

ConstantVector

Determinant

Diagonal

DiagonalMatrix

Dimension

Eigenvalues

Eigenvectors

Equal

GaussianElimination

GenerateEquations

GenerateMatrix

GramSchmidt

HermitianTranspose

Id

IdentityMatrix

IntersectionBasis

IsDefinite

IsOrthogonal

IsSimilar

IsUnitary

JordanBlockMatrix

JordanForm

LeastSquares

LinearSolve

LUDecomposition

MinimalPolynomial

Minor

MultiplyRow

Norm

Normalize

NullSpace

Pivot

QRDecomposition

RandomMatrix

RandomVector

Rank

ReducedRowEchelonForm

ReflectionMatrix

RotationMatrix

RowDimension

RowSpace

SumBasis

SwapRow

Trace

Transpose

UnitVector

VectorAngle

ZeroMatrix

ZeroVector

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To use hardware floating-point computations and treat symbols as complex-valued, use the SetDefault command in the (main) Student subpackage.  Local control is available for the complex-versus-real assumption by appropriate use of the conjugate option on relevant Student[LinearAlgebra] commands.  This local control is not available for the hardware-versus-software floating-point context.  These variations are illustrated in the following examples.

with(Student):

with(LinearAlgebra):

  

Normal usage:

Norm(<a,b>, 2);

a2&plus;b2

(1)
  

To assume that the symbols are complex for a particular computation:

Norm(<a,b>, 2, conjugate);

a2&plus;b2

(2)
  

To assume that symbols are complex in any computation:

SetDefault(conjugate = true);

conjugate&equals;false

(3)

Norm(<a,b>, 2);

a2&plus;b2

(4)
  

Normal floating-point computation:

<1.2,3.4> . <1.3,4.2>;

15.8399999999999999

(5)
  

For floating-point computations to take place in hardware whenever possible:

SetDefault(hardwarefloats=true);

hardwarefloats&equals;false

(6)

<1.2,3.4> . <1.3,4.2>;

15.8399999999999999

(7)
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There is special notation for the transpose and Hermitian transpose operations for Matrices and Vectors: M&plus; computes the transpose of M, where M is a Matrix or Vector (or scalar); and  M&ast; computes the Hermitian (conjugate) transpose of M.

A := <<a,b>|<c,d>|<e,f>>;

A:=acebdf

(8)

v := <a | b | c>;

v:=abc

(9)

A^+, v^+;

abcdef&comma;abc

(10)

A^*, v^*;

a&conjugate0;b&conjugate0;c&conjugate0;d&conjugate0;e&conjugate0;f&conjugate0;&comma;a&conjugate0;b&conjugate0;c&conjugate0;

(11)

See Also

LinearAlgebra Computation Example Worksheet, Student, Student[LinearAlgebra], Student[LinearAlgebra][InteractiveOverview], Student[LinearAlgebra][VisualizationOverview]


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