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English: Visual representation of a singular value decomposition (SVD) of the 2-dimensional real shearing

The upper left shows the unit disc in blue together with the two canonical unit vectors. The upper right shows the action of M on the unit disc: it distorts the circle to an ellipse. The SVD decomposes M into three simple transformations: a rotation V*, a scaling Σ along the coordinate axes and a second rotation U. The SVD reveals the lengths σ1 resp. σ2 of the semi-major axis resp. semi-minor axis of the ellispe; they are just the singular values which occur as diagonal elements of the scaling Σ. The rotation of the ellipse with respect to the coordinate axes is given by U. In this particular case the decomposition is as follows:

  • σ1 = Φ where Φ ≈ 1.618 denotes the golden ratio
  • σ2 = 1/Φ
  • V* = a clockwise rotation by α with tan(α) = Φ, i.e. V is a rotation by −α ≈ −58.28°.
  • U = a counter clockwise rotation by β where β satisfies tan(β) = Φ−1, i.e. β ≈ 31.72°.
Notice that Σ is unique, but V and U are not. We could add a rotation by 180° to both V and U or add some reflections.


Deutsch: Veranschaulichung einer Singulärwertzerlegung (SVD) der 2-dimensionalen, reellen Scherung

Oben links sieht man den Einheitskreis in blau zusammen mit den Standard-Einheitsvektoren. Oben rechts sieht man das Bild des Einheitskreises unter M: der Kreis wird zu einer Ellipse verzogen. Die SVD zerlegt M in drei einfache Transformationen: eine Rotation V*, eine Dehnung Σ entlang der Koordinatenachsen und eine zweite Rotation U. Die Zerlegung lässt direkt die Längen σ1 bzw. σ2 der großen bzw. kleinen Halbachse der Ellipse erkennen; die Werte stehen in der Hauptdiagonalen von Σ. Die Rotation der Ellipse in Bezug auf das Koordinatensystem wird durch U beschrieben.

In diesem speziellen Fall sieht die Singulärwertzerlegung aus wie folgt::

  • σ1 = Φ wobei Φ ≈ 1.618 den Goldenen Schnitt bezeichnet
  • σ2 = 1/Φ
  • V* = eine Drehung um α im Uhrzeigersinn, wobei tan(α) = Φ, d.h. V ist eine Drehung um −α ≈ −58.28°.
  • U = eine Drehung gegen den Uhrzeigersinn um β wobei gilt tan(β) = Φ−1, d.h. β ≈ 31.72°.
In der Zerlegung ist nur Σ eindeutig bestimmt; U und V sind es nicht: zu U und V könnten Drehungen um 180° oder Spiegelungen hinzugenommen werden.
Date
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Author Georg-Johann
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Illustration of the singular value decomposition UΣV* of a real 2×2 matrix M.

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28 August 2010

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Date/TimeThumbnailDimensionsUserComment
current06:40, 31 August 2010Thumbnail for version as of 06:40, 31 August 2010512 × 463 (12 KB)Georg-Johannfix rotation, cleanup svg
22:39, 30 August 2010Thumbnail for version as of 22:39, 30 August 2010512 × 463 (15 KB)Georg-Johann{{Information |Description={{en|1=Visualisation of a singular value decomposition. details will follow}} |Source={{own}} |Author=Georg-Johann |Date=2010-08-28 |Permission= |other_versions= }} Category:SVG mathematics

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