File:Singular-Value-Decomposition.svg

Summary

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

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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.

Copyright status:

GNU Free Documentation License, Version 1.2

Source:

https://commons.wikimedia.org/wiki/User:Georg-Johann