Singular value – Wikipedia

Square roots of the eigenvalues of the self-adjoint operator

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator

${displaystyle T:Xrightarrow Y}$

acting between Hilbert spaces

${displaystyle X}$

and

${displaystyle Y}$

, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator

${displaystyle T^{*}T}$

(where

${displaystyle T^{*}}$

denotes the adjoint of

${displaystyle T}$

).

The singular values are non-negative real numbers, usually listed in decreasing order (σ₁(T), σ₂(T), …). The largest singular value σ₁(T) is equal to the operator norm of T (see Min-max theorem).

If T acts on Euclidean space

${displaystyle mathbb {R} ^{n}}$

, there is a simple geometric interpretation for the singular values: Consider the image by

${displaystyle T}$

of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of

${displaystyle T}$

(the figure provides an example in

${displaystyle mathbb {R} ^{2}}$

).

The singular values are the absolute values of the eigenvalues of a normal matrix A, because the spectral theorem can be applied to obtain unitary diagonalization of

${displaystyle A}$

as

${displaystyle A=ULambda U^{*}}$

. Therefore,

${textstyle {sqrt {A^{*}A}}={sqrt {ULambda ^{*}Lambda U^{*}}}=Uleft|Lambda right|U^{*}}$

.

Most norms on Hilbert space operators studied are defined using s-numbers. For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence s-numbers are useful in classifying different operators.

In the finite-dimensional case, a matrix can always be decomposed in the form

${displaystyle mathbf {USigma V^{*}} }$

, where

${displaystyle mathbf {U} }$

and

${displaystyle mathbf {V^{*}} }$

are unitary matrices and

${displaystyle mathbf {Sigma } }$

is a rectangular diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.

Basic properties[edit]

For

${displaystyle Ain mathbb {C} ^{mtimes n}}$

, and

${displaystyle i=1,2,ldots ,min{m,n}}$

.

Min-max theorem for singular values. Here

${displaystyle U:dim(U)=i}$

is a subspace of

${displaystyle mathbb {C} ^{n}}$

of dimension

${displaystyle i}$

.

${displaystyle {begin{aligned}sigma _{i}(A)&=min _{dim(U)=n-i+1}max _{underset {|x|_{2}=1}{xin U}}left|Axright|_{2}.\sigma _{i}(A)&=max _{dim(U)=i}min _{underset {|x|_{2}=1}{xin U}}left|Axright|_{2}.end{aligned}}}$

Matrix transpose and conjugate do not alter singular values.

${displaystyle sigma _{i}(A)=sigma _{i}left(A^{textsf {T}}right)=sigma _{i}left(A^{*}right).}$

For any unitary

${displaystyle Uin mathbb {C} ^{mtimes m},Vin mathbb {C} ^{ntimes n}.}$

${displaystyle sigma _{i}(A)=sigma _{i}(UAV).}$

Relation to eigenvalues:

${displaystyle sigma _{i}^{2}(A)=lambda _{i}left(AA^{*}right)=lambda _{i}left(A^{*}Aright).}$

Relation to trace:

${displaystyle sum _{i=1}^{n}sigma _{i}^{2}={text{tr}} A^{ast }A}$

.

If

${displaystyle A^{top }A}$

is full rank, the product of singular values is

${displaystyle {sqrt {det A^{top }A}}}$

.

If

${displaystyle AA^{top }}$

is full rank, the product of singular values is

${displaystyle {sqrt {det AA^{top }}}}$

.

If

${displaystyle A}$

is full rank, the product of singular values is

${displaystyle |det A|}$

.

Inequalities about singular values[edit]

See also.^[1]

Singular values of sub-matrices[edit]

For

${displaystyle Ain mathbb {C} ^{mtimes n}.}$

1. Let
   ${displaystyle B}$

   denote

   ${displaystyle A}$

   with one of its rows or columns deleted. Then

   $${displaystyle sigma _{i+1}(A)leq sigma _{i}(B)leq sigma _{i}(A)}$$

   

2. Let
   ${displaystyle B}$

   denote

   ${displaystyle A}$

   with one of its rows and columns deleted. Then

   $${displaystyle sigma _{i+2}(A)leq sigma _{i}(B)leq sigma _{i}(A)}$$

   

3. Let
   ${displaystyle B}$

   denote an

   ${displaystyle (m-k)times (n-l)}$

   submatrix of

   ${displaystyle A}$

   . Then

   $${displaystyle sigma _{i+k+l}(A)leq sigma _{i}(B)leq sigma _{i}(A)}$$

   

Singular values of A + B[edit]

For

${displaystyle A,Bin mathbb {C} ^{mtimes n}}$

1. $${displaystyle sum _{i=1}^{k}sigma _{i}(A+B)leq sum _{i=1}^{k}(sigma _{i}(A)+sigma _{i}(B)),quad k=min{m,n}}$$

   

2. $${displaystyle sigma _{i+j-1}(A+B)leq sigma _{i}(A)+sigma _{j}(B).quad i,jin mathbb {N} , i+j-1leq min{m,n}}$$

   

Singular values of AB[edit]

For

${displaystyle A,Bin mathbb {C} ^{ntimes n}}$

1. $${displaystyle {begin{aligned}prod _{i=n}^{i=n-k+1}sigma _{i}(A)sigma _{i}(B)&leq prod _{i=n}^{i=n-k+1}sigma _{i}(AB)\prod _{i=1}^{k}sigma _{i}(AB)&leq prod _{i=1}^{k}sigma _{i}(A)sigma _{i}(B),\sum _{i=1}^{k}sigma _{i}^{p}(AB)&leq sum _{i=1}^{k}sigma _{i}^{p}(A)sigma _{i}^{p}(B),end{aligned}}}$$

   

2. $${displaystyle sigma _{n}(A)sigma _{i}(B)leq sigma _{i}(AB)leq sigma _{1}(A)sigma _{i}(B)quad i=1,2,ldots ,n.}$$

   

For

${displaystyle A,Bin mathbb {C} ^{mtimes n}}$

^[2]

Singular values and eigenvalues[edit]

For

${displaystyle Ain mathbb {C} ^{ntimes n}}$

.

1. See^[3]
   $${displaystyle lambda _{i}left(A+A^{*}right)leq 2sigma _{i}(A),quad i=1,2,ldots ,n.}$$

   

2. Assume
   ${displaystyle left|lambda _{1}(A)right|geq cdots geq left|lambda _{n}(A)right|}$

   . Then for

   ${displaystyle k=1,2,ldots ,n}$

   :

   1. Weyl’s theorem
      $${displaystyle prod _{i=1}^{k}left|lambda _{i}(A)right|leq prod _{i=1}^{k}sigma _{i}(A).}$$

      

   2. For
      ${displaystyle p>0}$

      

      $${displaystyle sum _{i=1}^{k}left|lambda _{i}^{p}(A)right|leq sum _{i=1}^{k}sigma _{i}^{p}(A).}$$