Gramian matrix

The Gramian matrix is also named after him.

It is a kind of Gramian matrix.

The Gramian matrix of any orthonormal basis is the identity matrix.

(The matrix containing in the location is known as the Gramian matrix.)

To have linear independency Gramian matrix have to be nonsingular, i.e., invertible.

In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is.

Mercer's theorem itself is a generalization of the result that any positive semidefinite matrix is the Gramian matrix of a set of vectors.

An matrix is said to be positive semidefinite if it is the gramian matrix of some vectors (i.e. if there exist vectors such that for all ).

Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp.

Further, in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix.

The Gramian matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors.

In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by .

To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the :