Wikipedia / Gram matrix
Wikipedia / Gram matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product. for the case that the vector coordinates are real numbers.
An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero.
It is named after Jørgen Pedersen Gram.
Examples
with the usual Euclidean dot product is its transpose. For complex is the conjugate transpose.
Given square-integrable functions is:
is the complex conjugate.
For any bilinear form on a finite-dimensional vector space over any field is symmetric.
Applications
In Riemannian geometry-dimensional Riemannian manifold for
If the vectors are centered random variables, the Gramian is approximately proportional to the covariance matrix, with the scaling determined by the number of elements in the vector.
In quantum chemistry, the Gram matrix of a set of basis vectors is the overlap matrix.
In control theory (or more generally systems theory), the controllability Gramian and observability Gramian determine properties of a linear system.
Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp.79–94).
In the finite element method, the Gram matrix arises from approximating a function from a finite dimensional space; the Gram matrix entries are then the inner products of the basis functions of the finite dimensional subspace.
In machine learning, kernel functions are often represented as Gram matrices. (Also see kernel PCA)
Since the Gram matrix over 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 inner 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).
Finding a vector realisation
, one can decompose it as:
,
is the conjugate transpose in the real case).
is the rank. Various ways to obtain such a decomposition include computing the Cholesky decomposition or taking the non-negative square root.
can be seen as n (or k, in the real case). Then
where the dot product.
Thus a Hermitian matrix. Such vectors are called a vector realization of The infinite-dimensional analog of this statement is Mercer's theorem.
Uniqueness of vector realizations
(any orthogonal transformation, that is, any Euclidean isometry orthogonal matrix is also
are unique up to orthogonal transformations and 0 to 0.
The same holds in the complex case, with unitary transformations then there is a unitary.
Other properties
is also a normal matrix.
The Gram matrix of any orthonormal basis is the identity matrix. In particular, 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.
equals the dimension of the space spanned by these vectors.
Gram determinant
The Gram determinant or Gramian
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 volume of the simplex formed by the vectors is .
.
. There exists a unique solution with minimal norm:The Gram determinant can also be expressed in terms of the exterior product of vectors by
The Gram determinant therefore supplies an inner product for the space . If an orthonormal basis ei, on is given, the vectors
will constitute an orthonormal basis of n-dimensional volumes on the space amounts to an n-dimensional Pythagorean theorem.
,
then the Gram determinant can be written as the difference of two Gram determinants,
-st dimension. Note that in the common case that , the second term on the right-hand side will be zero.
Constructing an orthonormal basis
, where the inner product is conjugate linear in the first argument, one can construct an orthonormal basis
.
. One can check that these new vectors are orthonormal:
.
See also
References
External links
Volumes of parallelograms by Frank Jones