the reals is a symmetric matrix, it is diagonalizable and its eigenvalues are non-negative. The diagonalization of the Gram matrix is the singular value decomposition. Properties Positive-semidefiniteness The Gram matrix is symmetric in the case the real product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation: The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of the inner-product, and the last from the positive definiteness of the inner product. Note that this also shows that the Gramian matrix is positive definite if and only if the vectors are linearly independent (that is, for all ). Finding a vector realization Given any positive semidefinite matrix , one can decompose it as: , where is the conjugate transpose of (or in the real case). Here is a matrix, where is the rank of . Various ways to obtain such a decomposition include computing the Cholesky decomposition or taking the non-negative square root of . The columns of can be seen as n vectors in (or k-dimensional Euclidean space , in the real case). Then where the dot product is the usual inner product on . Thus a Hermitian matrix is positive semidefinite if and only if it is the Gram matrix of some vectors . Such vectors are called a vector realization of The infinite-dimensional analog of this statement is Mercer's theorem. Uniqueness of vector realizations If is the Gram matrix of vectors in then applying any rotation or reflection of (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any orthogonal matrix , the Gram matrix of is also This is the only way in which two real vector realizations of can differ: the vectors are unique up to orthogonal transformations. In other words, the dot products and are equal if and only if some rigid transformation of transforms the vectors to and 0 to 0. The same holds in the complex case, with unitary transformations in place of orthogonal ones. That is, if the Gram matrix of vectors is equal to the Gram