**Today's Topics:**

Date: Fri, 22 May 87 12:27:31 EDT

From: stewart@thales.cs.umd.edu (G. W. Stewart)

To: na@score.stanford.edu

I am trying to find out who first used the term

singular value. As you can see from the attached

note, it comes from integral equations and was in

use as early as 1937. I would appreciate any help.

Pete Stewart

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Note on the Name\\

Singular Value Decomposition

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In a note in the Statistical Discussion Forum of

the {\it Journal of Statistical Planning and Inference}

I. J. Good (1986) objects to the name ``singular value

decomposition,'' preferring ``singular decomposition.''

Although the decomposition itself was discovered

independently by Beltrami (1873) and Jordan (1874),

and has since been frequently rediscovered, the name

singular value comes from the literature on integral

equations. In 1907 Erhard Schmidt introduced the

``eigenvalues''---the reciprocals of our singular values---

of an integral equation with a nonsymmetric kernal. Since

the name eigenvalue is obviously inapropriate for these

quantities, they came to be called singular values, though

I am uncertain of who first used the name. The earliest

reference I can find is Smithies (1937). It is interesting

to note that Schmidt proved the so-called Eckart-Young approximation

theorem (1936) in its full generality for integral operators,

and his name should be associated with the theorem.

Since the term singular value is well established,

there is no good reason not to use it attributively

to describe a decomposition that exhibits singular values.

Against the name singular decomposition, it can be objected that

the word singular has many uses in and out of mathematics. For

example, the singular value decomposition is important, but is

it singularly important?

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References

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Beltrami, E. (1873)\\

``Sulle Funzioni Bilineari,''

{\it Giornale di Matematiche ud uso Degli Studenti Delle Universit\`a

Italiane} {\bf 11,} 98-106.

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Eckart, C. and G. Young (1936)\\

``The Approximation of One Matrix by Another of Lower Rank,''

{\it Psychometrika} {\bf 1,} 211-218.

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Good, I. J. (1986)\\

``F1. The Singular Decomposition of a Matrix: a Point of Terminology,''

{\it Journal of Statistical Planning and Inference} {\bf 14,} 411-412.

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Jordan, C. (1874)\\

``M\'emoire sur les formes bilin\'eaires,''

{\it Journal de Math\'ematiques Pures et Appliqu\'ees, Deuxi\'eme S\'erie}

{\bf 19,} 35-54.

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Schmidt, E. (1907)\\

``Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Tiel.

Entwicklung willk\"urlichen Fuuktionen nach System vorgeschriebener,''

{\it Mathematische Annalen} {\bf 63,} 433-476.

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Smithies, F. (1937)\\

``The Eigen-values and Singular Values of Integral Equations,''

{\it Proceedings of the London Mathematical Society} {\bf 43,}

255-279.

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