Singular Value

What is a Singular Value?

Singular value or s-number is a mathematical construct that is used in matrix algebra and elliptic integrals. In this article, we will only describe singular values within the context of linear algebra.

Mathematically, singular values can be defined as:

Given a square matrix A and its conjugate transpose AH, the conjugate values of A are the square roots of the eigenvalues of AH.

According to Steward (1993), the name “singular value” derives from the literature on integral equations. This concept was first defined by Baltic German mathematician Erhard Schmidt in 1907, who called them “eigenvalues”. Later, in 1937, the mathematician F. Smithies used the term singular values for the first time. And, in 1969, Gohberg and Krein provided another name for singular values when they translated this concept from Russian as “s-numbers”.

 

Why are Singular Values Important

Singular values are at the core of the Singular Value Decomposition method that allows a complex matrix A to be factorized into three different components:

Where U and V are unitary matrices and D is a diagonal matrix whose elements are the singular values of A.

This decomposition is very useful, as it translates a matrix that is difficult to manage into three matrices that are easier to manipulate.

 

Singular Values + LogicPlum

Singular values are an important part of linear algebra, and therefore of machine learning. LogicPlum’s platform applies them in many calculations used by the system to obtain an optimal model.

Although linear algebra requires specialized knowledge, LogicPlum’s users don’t need to master this area of mathematics to benefit from the models estimated by the platform. This is because all operations are done by the system in an automated manner. This capacity of the platform allows for all stakeholders to take part in the modeling process and to contribute according to their corresponding areas of expertise.

 

Additional Resources

For those who are interested in Smithies’ original paper:

Smithies, F. (1938) The Eigen‐Values and Singular Values of Integral Equations. Available at

https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-43.4.255

For those who enjoy reading about the history of mathematics:

Steward, G.W. (1993). On the Early History of the Singular Value Decomposition. Available at

https://www.math.ucdavis.edu/~saito/courses/229A/stewart-svd.pdf

For those wanting to explore more:

Weisstein, E.W. Singular Value. MathWorld–A Wolfram Web Resource. Available at https://mathworld.wolfram.com/SingularValue.html

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