matrix of vectors in then there is a unitary matrix (meaning ) such that for . Other properties Because , it is necessarily the case that and commute. That is, a real or complex Gram matrix is also a normal matrix. The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix. The rank of the Gram matrix of vectors in or equals the dimension of the space spanned by these vectors. Gram determinant The Gram determinant or Gramian is the determinant of the Gram matrix: If are vectors in then it is the square of the n-dimensional volume of the parallelotope formed by the vectors. In particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular. When the determinant and volume are zero. When , this reduces to the standard theorem that the absolute value of the determinant of n n-dimensional vectors is the n-dimensional volume. The Gram determinant is also useful for computing the volume of the simplex formed by the vectors; its volume is . The Gram determinant can also be expressed in terms of the exterior product of vectors by When the vectors are defined from the positions of points relative to some reference point , then the Gram determinant can be written as the difference of two Gram determinants, where each is the corresponding point supplemented with the coordinate value of 1 for an -st dimension. Note that in the common case that , the second term on the right-hand side will be zero. Constructing an orthonormal basis Given a set of linearly independent vectors with Gram matrix , one can construct an orthonormal basis The positive definite matrix is guaranteed to exist because, as mentioned above, the are linearly independent if and only if G is invertible and hence is positive definite (not just semidefinite). The inverse of positive definite matrix G is unique and also positive definite and thus has a unique positive definite square root . One can check that these new vectors are orthonormal: See also Controllability Gramian Observability Gramian References External