File:Diagonalization as rotation.gif
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Diagonalization_as_rotation.gif (357 × 197 pixels, file size: 556 KB, MIME type: image/gif, looped, 153 frames)
Summary
| Description |
English: You can visualize a matrix diagonalization as a rotation of your axis to align them with the matrix eigenvectors. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1192396841381515264 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
| Camera location |  | View this and other nearby images on: OpenStreetMap |  |
|---|
Mathematica 11.0 code
m = RandomReal[{-10, 10}, {3, 3}];
m = Round[(m + Transpose[m])/2, 0.01];
\[Lambda] = Eigenvalues[m];
\[Psi] = Eigenvectors[m];
angle[u_, v_] := ArcCos[u.v/(Norm[u] Norm[v])];
xyz = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
\[Alpha]1 = \[Alpha] /. NMinimize[Norm[EulerMatrix[{\[Alpha], 0, 0}].xyz[[2]] - Cross[{0, 0, 1}, \[Psi][[3]]]/Norm[Cross[{0, 0, 1}, \[Psi][[3]]]] ], \[Alpha]][[2]];(*rotate y in the plane perpendicular to Subscript[\[Psi], z]*)
\[Beta]1 = angle[{0, 0, 1}, \[Psi][[3]]];
\[Gamma]1 = -angle[Cross[{0, 0, 1}, \[Psi][[3]]], \[Psi][[2]]];
p1 = Table[
Grid[{{
PaddedForm[MatrixForm[Inverse[EulerMatrix[{\[Alpha], 0, 0}]].m.EulerMatrix[{\[Alpha], 0, 0}] ], {3, 2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}], Sphere[{0, 0, 0}, 0.1]
, Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha], 0, 0}].xyz[[j]]}], {j, 1, 3}] }, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
}}]
, {\[Alpha], 0, \[Alpha]1, \[Alpha]1/40}];
p2 = Table[
Grid[{{
PaddedForm[MatrixForm[Round[Inverse[EulerMatrix[{\[Alpha]1, \[Beta], 0}]].m.EulerMatrix[{\[Alpha]1, \[Beta], 0}], 0.01] ], {3,2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}],
Sphere[{0, 0, 0}, 0.1], Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha]1, \[Beta], 0}].xyz[[j]]}], {j, 1, 3}]
}, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
}}]
, {\[Beta], 0, \[Beta]1, \[Beta]1/40}];
p3 = Table[
Grid[{{
PaddedForm[MatrixForm[Round[Inverse[EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}]].m.EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}], 0.01] ], {3, 2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}], Sphere[{0, 0, 0}, 0.1], Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}].xyz[[j]]}], {j, 1, 3}]
}, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
}}]
, {\[Gamma], 0, \[Gamma]1, \[Gamma]1/40}];
ListAnimate[Join[p1, p2, p3, Table[p3[[-1]], 30]]]
Licensing
I, the copyright holder of this work, hereby publish it under the following licence:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighbouring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:31, 11 November 2019 | | 357 × 197 (556 KB) | wikimediacommons>Berto | User created page with UploadWizard |
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