Editing Gram matrix (section)

===Finding a vector realization===
{{See also|Positive definite matrix#Decomposition}}
Given any positive semidefinite matrix <math>M</math>, one can decompose it as:
: <math>M = B^\dagger B</math>,

where <math>B^\dagger</math> is the [[conjugate transpose]] of <math>B</math> (or <math>M = B^\textsf{T} B</math> in the real case).

Here <math>B</math> is a <math>k \times n</math> matrix, where <math>k</math> is the [[matrix rank|rank]] of <math>M</math>. Various ways to obtain such a decomposition include computing the [[Cholesky decomposition]] or taking the [[square root of a matrix|non-negative square root]] of <math>M</math>.

The columns <math>b^{(1)}, \dots, b^{(n)}</math> of <math>B</math> can be seen as ''n'' vectors in <math>\mathbb{C}^k</math> (or ''k''-dimensional Euclidean space <math>\mathbb{R}^k</math>, in the real case). Then
: <math>M_{ij} = b^{(i)} \cdot b^{(j)}</math>

where the [[dot product]] <math display="inline">a \cdot b = \sum_{\ell=1}^k a_\ell^* b_\ell</math> is the usual inner product on <math>\mathbb{C}^k</math>.

Thus a [[Hermitian matrix]] <math>M</math> is positive semidefinite if and only if it is the Gram matrix of some vectors <math>b^{(1)}, \dots, b^{(n)}</math>. Such vectors are called a '''vector realization''' of {{nowrap|<math>M</math>.}} The infinite-dimensional analog of this statement is [[Mercer's theorem]].