##  [Singular Value Decomposition](/singular-value-decomposition) 

  ##  [Singular Value Decomposition](https://natural.quantumdictionary.io/singular-value-decomposition-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A factorization of a finite-dimensional linear map (or matrix) A as A = U Σ V^*, where U and V are unitary (or orthogonal) matrices and Σ is a diagonal matrix with nonnegative real entries called singular values.

 

 

 

 

 





 

 



 ##  [Singular Value Decomposition](https://mathlogic.quantumdictionary.io/singular-value-decomposition-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A factorization of an m×n matrix A (real or complex) as A = U Σ V*, where U and V are unitary (orthogonal in the real case) matrices and Σ is a diagonal (rectangular) matrix with nonnegative real entries called singular values; it encodes the action of A on orthogonal directions.

 

 

 

 

 





 

 



 ##  [Singular Value Decomposition](https://algebra.quantumdictionary.io/singular-value-decomposition-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A factorization of a matrix A into A = U Σ V^* where U and V are unitary (orthogonal in real case) matrices and Σ is a diagonal (rectangular) matrix of nonnegative singular values, exposing rank, range, and nullspace structure.