Making an interactive visualization of the eigenvectors of two-dimensional matrices

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I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.

How can this sort of visualization be realized with Mathematica, leveraging its dynamical capabilities?

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

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up vote
2
down vote

favorite

I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.

How can this sort of visualization be realized with Mathematica, leveraging its dynamical capabilities?

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

add a commentÂ |Â

up vote
2
down vote

favorite

I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.

How can this sort of visualization be realized with Mathematica, leveraging its dynamical capabilities?

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.

How can this sort of visualization be realized with Mathematica, leveraging its dynamical capabilities?

graphics dynamic visualization eigenvalues eventhandler

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

edited 1 hour ago

asked 1 hour ago

glS

4,73911140

asked 1 hour ago

glS

4,73911140

asked 1 hour ago

glS

4,73911140

add a commentÂ |Â

1 Answer
1

active

oldest

votes

up vote
4
down vote

The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.

The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.

Here is the full code:

arrowRepresentationActionMatrix[matrix_] := 
 With[pts = MeshCoordinates@DiscretizeRegion@Region@Circle,
 With[finalPts = Dot[matrix, #] & /@ pts,
 Graphics[
 PointSize@0.012, Point@finalPts,
 Arrow /@ Thread@pts, finalPts
 ]
 ]];

DynamicModule[v1 = 0.7, -0.6, v2 = 0.6, 0.6, v3 = 1, 1, 
 movingPointIndex, matrix, frameSize = 1.5, numOfIterations = 30,
 Row[
 EventHandler[
 Dynamic[
 matrix = Transpose@v1, v2;
 Show[
 arrowRepresentationActionMatrix@matrix,
 Graphics[
 PointSize@0.02, Circle, Point@0, 0,
 Red, 
 If[TrueQ[movingPointIndex == 1], PointSize@0.04, 
 PointSize@0.03],
 Point@v1, Arrow@0, 0, v1,
 Green, 
 If[TrueQ[movingPointIndex == 2], PointSize@0.04, 
 PointSize@0.03],
 Point@v2, Arrow@0, 0, v2
 ,
 Dynamic@Blue, PointSize@0.03, Point@v3, Arrowheads@0.02,

 Arrow /@ 
 Partition[NestList[Dot[matrix, #] &, v3, numOfIterations], 
 2, 1]
 ,

 Dynamic@With[eigs = Eigenvectors@N@matrix,

 If[MatchQ[eigs, __Real ..], Purple, Thickness@0.01, 
 InfiniteLine@-#, # & /@ eigs, ]
 ]
 
 ],
 Frame -> True, 
 PlotRange -> Dynamic[-#, #, -#, # &@frameSize], 
 ImageSize -> 500
 ]
 ],
 "MouseDown" :> With[mp = MousePosition["Graphics"],
 movingPointIndex = 
 Position[v1, v2, v3, First@Nearest[v1, v2, v3, mp]][[1, 1]]
 ],
 "MouseDragged" :> ReleaseHold[
 Hold[Set][Hold[v1, v2, v3][[movingPointIndex]], 
 MousePosition["Graphics"]]
 ]
 ],
 Column@
 "PlotRange",
 VerticalSlider[Dynamic@frameSize, 1, 10, 0.01, 
 Appearance -> "Labeled"]
 , " ",
 Column@
 "Number of iterations",
 VerticalSlider[Dynamic@numOfIterations, 1, 40, 1, 
 Appearance -> "Labeled"]
 
 ]
 ]

and this is the result:

enter image description here

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

This is really cool
â€“Â user6014
1 hour ago

add a commentÂ |Â

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1 Answer
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active

oldest

votes

1 Answer
1

active

oldest

votes

up vote
4
down vote

The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.

The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.

Here is the full code:

arrowRepresentationActionMatrix[matrix_] := 
 With[pts = MeshCoordinates@DiscretizeRegion@Region@Circle,
 With[finalPts = Dot[matrix, #] & /@ pts,
 Graphics[
 PointSize@0.012, Point@finalPts,
 Arrow /@ Thread@pts, finalPts
 ]
 ]];

DynamicModule[v1 = 0.7, -0.6, v2 = 0.6, 0.6, v3 = 1, 1, 
 movingPointIndex, matrix, frameSize = 1.5, numOfIterations = 30,
 Row[
 EventHandler[
 Dynamic[
 matrix = Transpose@v1, v2;
 Show[
 arrowRepresentationActionMatrix@matrix,
 Graphics[
 PointSize@0.02, Circle, Point@0, 0,
 Red, 
 If[TrueQ[movingPointIndex == 1], PointSize@0.04, 
 PointSize@0.03],
 Point@v1, Arrow@0, 0, v1,
 Green, 
 If[TrueQ[movingPointIndex == 2], PointSize@0.04, 
 PointSize@0.03],
 Point@v2, Arrow@0, 0, v2
 ,
 Dynamic@Blue, PointSize@0.03, Point@v3, Arrowheads@0.02,

 Arrow /@ 
 Partition[NestList[Dot[matrix, #] &, v3, numOfIterations], 
 2, 1]
 ,

 Dynamic@With[eigs = Eigenvectors@N@matrix,

 If[MatchQ[eigs, __Real ..], Purple, Thickness@0.01, 
 InfiniteLine@-#, # & /@ eigs, ]
 ]
 
 ],
 Frame -> True, 
 PlotRange -> Dynamic[-#, #, -#, # &@frameSize], 
 ImageSize -> 500
 ]
 ],
 "MouseDown" :> With[mp = MousePosition["Graphics"],
 movingPointIndex = 
 Position[v1, v2, v3, First@Nearest[v1, v2, v3, mp]][[1, 1]]
 ],
 "MouseDragged" :> ReleaseHold[
 Hold[Set][Hold[v1, v2, v3][[movingPointIndex]], 
 MousePosition["Graphics"]]
 ]
 ],
 Column@
 "PlotRange",
 VerticalSlider[Dynamic@frameSize, 1, 10, 0.01, 
 Appearance -> "Labeled"]
 , " ",
 Column@
 "Number of iterations",
 VerticalSlider[Dynamic@numOfIterations, 1, 40, 1, 
 Appearance -> "Labeled"]
 
 ]
 ]

and this is the result:

enter image description here

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

This is really cool
â€“Â user6014
1 hour ago

add a commentÂ |Â

up vote
4
down vote

The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.

The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.

Here is the full code:

arrowRepresentationActionMatrix[matrix_] := 
 With[pts = MeshCoordinates@DiscretizeRegion@Region@Circle,
 With[finalPts = Dot[matrix, #] & /@ pts,
 Graphics[
 PointSize@0.012, Point@finalPts,
 Arrow /@ Thread@pts, finalPts
 ]
 ]];

DynamicModule[v1 = 0.7, -0.6, v2 = 0.6, 0.6, v3 = 1, 1, 
 movingPointIndex, matrix, frameSize = 1.5, numOfIterations = 30,
 Row[
 EventHandler[
 Dynamic[
 matrix = Transpose@v1, v2;
 Show[
 arrowRepresentationActionMatrix@matrix,
 Graphics[
 PointSize@0.02, Circle, Point@0, 0,
 Red, 
 If[TrueQ[movingPointIndex == 1], PointSize@0.04, 
 PointSize@0.03],
 Point@v1, Arrow@0, 0, v1,
 Green, 
 If[TrueQ[movingPointIndex == 2], PointSize@0.04, 
 PointSize@0.03],
 Point@v2, Arrow@0, 0, v2
 ,
 Dynamic@Blue, PointSize@0.03, Point@v3, Arrowheads@0.02,

 Arrow /@ 
 Partition[NestList[Dot[matrix, #] &, v3, numOfIterations], 
 2, 1]
 ,

 Dynamic@With[eigs = Eigenvectors@N@matrix,

 If[MatchQ[eigs, __Real ..], Purple, Thickness@0.01, 
 InfiniteLine@-#, # & /@ eigs, ]
 ]
 
 ],
 Frame -> True, 
 PlotRange -> Dynamic[-#, #, -#, # &@frameSize], 
 ImageSize -> 500
 ]
 ],
 "MouseDown" :> With[mp = MousePosition["Graphics"],
 movingPointIndex = 
 Position[v1, v2, v3, First@Nearest[v1, v2, v3, mp]][[1, 1]]
 ],
 "MouseDragged" :> ReleaseHold[
 Hold[Set][Hold[v1, v2, v3][[movingPointIndex]], 
 MousePosition["Graphics"]]
 ]
 ],
 Column@
 "PlotRange",
 VerticalSlider[Dynamic@frameSize, 1, 10, 0.01, 
 Appearance -> "Labeled"]
 , " ",
 Column@
 "Number of iterations",
 VerticalSlider[Dynamic@numOfIterations, 1, 40, 1, 
 Appearance -> "Labeled"]
 
 ]
 ]

and this is the result:

enter image description here

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

This is really cool
â€“Â user6014
1 hour ago

add a commentÂ |Â

up vote
4
down vote

The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.

The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.

Here is the full code:

arrowRepresentationActionMatrix[matrix_] := 
 With[pts = MeshCoordinates@DiscretizeRegion@Region@Circle,
 With[finalPts = Dot[matrix, #] & /@ pts,
 Graphics[
 PointSize@0.012, Point@finalPts,
 Arrow /@ Thread@pts, finalPts
 ]
 ]];

DynamicModule[v1 = 0.7, -0.6, v2 = 0.6, 0.6, v3 = 1, 1, 
 movingPointIndex, matrix, frameSize = 1.5, numOfIterations = 30,
 Row[
 EventHandler[
 Dynamic[
 matrix = Transpose@v1, v2;
 Show[
 arrowRepresentationActionMatrix@matrix,
 Graphics[
 PointSize@0.02, Circle, Point@0, 0,
 Red, 
 If[TrueQ[movingPointIndex == 1], PointSize@0.04, 
 PointSize@0.03],
 Point@v1, Arrow@0, 0, v1,
 Green, 
 If[TrueQ[movingPointIndex == 2], PointSize@0.04, 
 PointSize@0.03],
 Point@v2, Arrow@0, 0, v2
 ,
 Dynamic@Blue, PointSize@0.03, Point@v3, Arrowheads@0.02,

 Arrow /@ 
 Partition[NestList[Dot[matrix, #] &, v3, numOfIterations], 
 2, 1]
 ,

 Dynamic@With[eigs = Eigenvectors@N@matrix,

 If[MatchQ[eigs, __Real ..], Purple, Thickness@0.01, 
 InfiniteLine@-#, # & /@ eigs, ]
 ]
 
 ],
 Frame -> True, 
 PlotRange -> Dynamic[-#, #, -#, # &@frameSize], 
 ImageSize -> 500
 ]
 ],
 "MouseDown" :> With[mp = MousePosition["Graphics"],
 movingPointIndex = 
 Position[v1, v2, v3, First@Nearest[v1, v2, v3, mp]][[1, 1]]
 ],
 "MouseDragged" :> ReleaseHold[
 Hold[Set][Hold[v1, v2, v3][[movingPointIndex]], 
 MousePosition["Graphics"]]
 ]
 ],
 Column@
 "PlotRange",
 VerticalSlider[Dynamic@frameSize, 1, 10, 0.01, 
 Appearance -> "Labeled"]
 , " ",
 Column@
 "Number of iterations",
 VerticalSlider[Dynamic@numOfIterations, 1, 40, 1, 
 Appearance -> "Labeled"]
 
 ]
 ]

and this is the result:

enter image description here

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.

The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.

Here is the full code:

arrowRepresentationActionMatrix[matrix_] := 
 With[pts = MeshCoordinates@DiscretizeRegion@Region@Circle,
 With[finalPts = Dot[matrix, #] & /@ pts,
 Graphics[
 PointSize@0.012, Point@finalPts,
 Arrow /@ Thread@pts, finalPts
 ]
 ]];

DynamicModule[v1 = 0.7, -0.6, v2 = 0.6, 0.6, v3 = 1, 1, 
 movingPointIndex, matrix, frameSize = 1.5, numOfIterations = 30,
 Row[
 EventHandler[
 Dynamic[
 matrix = Transpose@v1, v2;
 Show[
 arrowRepresentationActionMatrix@matrix,
 Graphics[
 PointSize@0.02, Circle, Point@0, 0,
 Red, 
 If[TrueQ[movingPointIndex == 1], PointSize@0.04, 
 PointSize@0.03],
 Point@v1, Arrow@0, 0, v1,
 Green, 
 If[TrueQ[movingPointIndex == 2], PointSize@0.04, 
 PointSize@0.03],
 Point@v2, Arrow@0, 0, v2
 ,
 Dynamic@Blue, PointSize@0.03, Point@v3, Arrowheads@0.02,

 Arrow /@ 
 Partition[NestList[Dot[matrix, #] &, v3, numOfIterations], 
 2, 1]
 ,

 Dynamic@With[eigs = Eigenvectors@N@matrix,

 If[MatchQ[eigs, __Real ..], Purple, Thickness@0.01, 
 InfiniteLine@-#, # & /@ eigs, ]
 ]
 
 ],
 Frame -> True, 
 PlotRange -> Dynamic[-#, #, -#, # &@frameSize], 
 ImageSize -> 500
 ]
 ],
 "MouseDown" :> With[mp = MousePosition["Graphics"],
 movingPointIndex = 
 Position[v1, v2, v3, First@Nearest[v1, v2, v3, mp]][[1, 1]]
 ],
 "MouseDragged" :> ReleaseHold[
 Hold[Set][Hold[v1, v2, v3][[movingPointIndex]], 
 MousePosition["Graphics"]]
 ]
 ],
 Column@
 "PlotRange",
 VerticalSlider[Dynamic@frameSize, 1, 10, 0.01, 
 Appearance -> "Labeled"]
 , " ",
 Column@
 "Number of iterations",
 VerticalSlider[Dynamic@numOfIterations, 1, 40, 1, 
 Appearance -> "Labeled"]
 
 ]
 ]

and this is the result:

enter image description here

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

edited 1 hour ago

answered 1 hour ago

glS

4,73911140

answered 1 hour ago

glS

4,73911140

answered 1 hour ago

glS

4,73911140

This is really cool
â€“Â user6014
1 hour ago

add a commentÂ |Â

This is really cool
â€“Â user6014
1 hour ago

This is really cool
â€“Â user6014
1 hour ago

add a commentÂ |Â

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