|
In systems theory and linear algebra, a Gramian matrix is a real-valued symmetric matrix that can be used to test for linear independence of functions. The Gramian matrix of a set of functions  is defined as: Systems theory is a transdisciplinary/multiperspectual theory that studies structure and properties of systems in terms of relationships from which new properties of wholes emerge. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...
In mathematics, the real numbers may be described informally in several different ways. ...
In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ...
Partial plot of a function f. ...
![G=[G_{ij}],,,G_{ij}=int_{t_0}^{t_f} l_i(tau)l_j(tau), dtau](http://upload.wikimedia.org/math/7/e/c/7ec106189315eb50bbcd840684ae9d8e.png) The functions are linearly independent if and only if G is nonsingular. Its determinant is known as the Gram determinant or Gramian. It is named for Jørgen Pedersen Gram. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
Jørgen Pedersen Gram (June 27, 1850 - April 29, 1916) was a Danish mathematician who was born in Nustrup, Denmark and died in Copenhagen, Denmark. ...
In fact this is a special case of a quantitative measure of linear independence of vectors, available in any Hilbert space. According to that definition, for E a real prehilbert space, if In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
 are n vectors of E, the associated Gram matrix is the symmetric matrix  . The Gram determinant is the determinant of this matrix,  All eigenvalues of a Gramian matrix are real and non-negative and the matrix is thus also positive semidefinite. In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...
See also
The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...
The Observability Gramian is a Gramian used in optimal control theory to determine whether or not a linear system is observable. ...
External links - An application of the Gramian matrix in general linear algebra
|
Feeds |
Contact