Gram matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors 
 
 
 
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 
 
 {\displaystyle v_{1},\dots ,v_{n}}
 
 in an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> is the <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> of <a href="/facts/Inner_product/HoyjElEy">inner products</a>, whose entries are given by the <a href="/facts/Inner_product/HoyjElEy">inner product</a> 
 
 
 
 
 G
 
 i
 j
 
 
 =
 
 ⟨
 
 
 v
 
 i
 
 
 ,
 
 v
 
 j
 
 
 
 ⟩
 
 
 
 {\displaystyle G_{ij}=\left\langle v_{i},v_{j}\right\rangle }
 
. If the vectors 
 
 
 
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 
 
 {\displaystyle v_{1},\dots ,v_{n}}
 
 are the columns of matrix 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the Gram matrix is 
 
 
 
 
 X
 
 †
 
 
 X
 
 
 {\displaystyle X^{\dagger }X}
 
 in the general case that the vector coordinates are complex numbers, which simplifies to 
 
 
 
 
 X
 
 ⊤
 
 
 X
 
 
 {\displaystyle X^{\top }X}
 
 for the case that the vector coordinates are real numbers.
An important application is to compute <a href="/facts/Linear_independence/8JrHfIIa">linear independence</a>: a set of vectors are linearly independent if and only if the Gram determinant (the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the Gram matrix) is non-zero.
It is named after <a href="/facts/J%C3%B8rgen_Pedersen_Gram/r6Hodn4n">Jørgen Pedersen Gram</a>.

Gram matrix open-in-new

Gram matrix