Questions tagged [singular-values]

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator $~T : X → Y~$ acting between Hilbert spaces $~X~$ and $~Y~$ , are the square roots of non-negative eigenvalues of the self-adjoint operator $~T'~T~$  (where $~T'~$ denotes the adjoint of $~T~$).

Definition: Let $~A~$ be an $~m × n~$ matrix and $~λ_1,λ_2,~\cdots~, λ_n~$ denote the eigenvalues of $~A^{\text{T}}~ A~$, with repetitions. Order these so that $~λ_1 ≥ λ_2 ≥ \cdots ≥ λ_n ≥ 0~$. Let $~σ_i = \sqrt{λ_i~}~$, so that $~σ_1 ≥ σ_2 ≥\cdots ≥ σ_n ≥ 0~$. The numbers $~σ_1,~ σ_2 ,~\cdots ,~ σ_n~$ are called singular values of $~A~$.

Singular values play an important role where the matrix is a transformation from one vector space to a different vector space, possibly with a different dimension.

• The number of nonzero singular values of $~A~$ equals the rank of $~A~$.
• In particular, if $~A~$ is an $~m × n~$ matrix with $~m < n~$ , then $~A~$ has at most $~m~$ nonzero singular values, because rank$~(A) ≤ m~$.
• Let $~A~$ be an $~m × n~$ matrix. Then the maximum value of $~||Ax||~$, where $~x~$ ranges over unit vectors in $~\mathbb(R)^n~$, is the largest singular value $~σ_1~$, and this is achieved when $~x~$ is an eigenvector of$~A^{\text{T}}~ A~$with eigenvalue $~\sigma_1^2~$. (This is the geometric significance of singular values.)

References:

https://en.wikipedia.org/wiki/Singular_value

http://mathworld.wolfram.com/SingularValue.html