Quelle Sweep.g
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#! @Chapter Examples and Tests
#! @Section Sweep
#! $\href{https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-graph/}{\textrm{Geometric interpretation of sweeping a matrix by Terence Tao.}}$
LoadPackage( "LinearAlgebraForCAP" );
LoadPackage( "GeneralizedMorphismsForCAP" );
#! @Example
Q := HomalgFieldOfRationals();;
V := VectorSpaceObject( 3, Q );;
mat := HomalgMatrix( [ [ 9, 8, 7 ], [ 6, 5, 4 ], [ 3, 2, 1 ] ], 3, 3, Q );;
alpha := VectorSpaceMorphism( V, mat, V );;
graph := FiberProductEmbeddingInDirectSum(
[ alpha, IdentityMorphism( V ) ] );;
Display( graph );
#! [ [ 1, -2, 1, 0, 0, 0 ],
#! [ -4/3, 7/3, 0, 2, 1, 0 ],
#! [ 5/3, -8/3, 0, -1, 0, 1 ] ]
#!
#! A morphism in Category of matrices over Q
D := DirectSum( V, V );;
rotmat := HomalgMatrix( [ [ 0, 0, 0, -1, 0, 0 ],
[ 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0 ],
[ 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 1 ] ],
6, 6, Q );;
rot := VectorSpaceMorphism( D, rotmat, D );;
p := PreCompose( graph, rot );;
Display( p );
#! [ [ 0, -2, 1, -1, 0, 0 ],
#! [ 2, 7/3, 0, 4/3, 1, 0 ],
#! [ -1, -8/3, 0, -5/3, 0, 1 ] ]
#!
#! A morphism in Category of matrices over Q
pi1 := ProjectionInFactorOfDirectSum( [ V, V ], 1 );;
pi2 := ProjectionInFactorOfDirectSum( [ V, V ], 2 );;
reversed_arrow := PreCompose( p, pi1 );;
arrow := PreCompose( p, pi2 );;
g := GeneralizedMorphismBySpan( reversed_arrow, arrow );;
IsHonest( g );
#! true
sweep_1_alpha := HonestRepresentative( g );;
Display( sweep_1_alpha );
#! [ [ -1/9, 8/9, 7/9 ],
#! [ 2/3, -1/3, -2/3 ],
#! [ 1/3, -2/3, -4/3 ] ]
#!
#! A morphism in Category of matrices over Q
Display( alpha );
#! [ [ 9, 8, 7 ],
#! [ 6, 5, 4 ],
#! [ 3, 2, 1 ] ]
#!
#! A morphism in Category of matrices over Q
#! @EndExample
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2026-03-28
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