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After a linear transformation, some vectors may not change direction, they only scale by a number. The scaling factor of those vectors is called eigenvalue.

Can we think of singular values in this manner? Since eigenvalues are related to only square matrices, Are singular values the generalized "scaling factors" of a linear transformation (matrix)?

Thank you.

Ardhendu
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  • they're the square roots of the eigenvalues of $A^TA$. – Jürgen Sukumaran Sep 05 '23 at 14:20
  • See simple geometric interpretation for the singular values – Ricky Sep 05 '23 at 14:20
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    From the Wikipedia article: "If $T$ acts on Euclidean space there is a simple geometric interpretation for the singular values: Consider the image by $T$ of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of $T$ " – whpowell96 Sep 05 '23 at 14:22
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    Hint: The singular values represent the lengths of the semiaxes of the ellipsoid that the matrix maps the unit sphere to in the input space. They describe how the matrix scales and rotates vectors without specifying any fixed set of directions. – MathRookie2204 Sep 05 '23 at 14:23
  • @mordecaiiwazuki So if we take the square root of the eigenvalues of $A^TA$, what does it have to do with the matrix $A$ itself? – Ardhendu Sep 05 '23 at 14:27
  • @MathRookie2204 so can we say that singular values are also like scaling factors of a transformation? – Ardhendu Sep 05 '23 at 14:30
  • @Ricky I got your point, but can we say singular values are the generalization of eigenvalues? – Ardhendu Sep 05 '23 at 14:34
  • See a very good answer here – Jean Marie Sep 05 '23 at 14:35
  • Actually, I am tempted to say singular values are a general form of eigenvalues. Is this true? If so, provide some intuition relating them. – Ardhendu Sep 05 '23 at 14:39
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    @Ardhendu Eigenvalues tell you something about how the map behaves along a line. Specifically, it tells you that a line is preserved under the transformation, by stretching shrinking, or reflecting it onto itself. Singular values tell you what shape the unit sphere is morphed into under the transformation. It doesn't tell you which subspaces are invariant, but it does say something more holistic about the transformation. They are different to eigenvalues, but they certainly are related conceptually (indeed, they are eigenvalues of a different, related map). – Theo Bendit Sep 05 '23 at 15:22
  • +1 @TheoBendit why don't you polish/elaborate your comment a little bit and post it as an answer? I guess this is what I was looking for. – Ardhendu Sep 05 '23 at 17:15
  • Is it true that the transformation is stretched along the eigen vectors of left singular vectors? – Ardhendu Sep 06 '23 at 06:37
  • @Ardhendu No, only right eigenvectors. Where right eigenvectors span lines stablilised by the transformation, left eigenvectors are normal to hyperplanes stabilised by the transformation. – Theo Bendit Sep 07 '23 at 16:32

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