GRAMIAN MATRIX

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In systems theory and linear algebra, the 'Gramian matrix' of a set of functions $\{l\_i(cdot),,i=1,dots,n\}$ is a real-valued symmetric matrix
$G=[G\_\{ij\}]$, where $G\_\{ij\}=int\_\{t\_0\}^\{t\_f\}\; l\_i(\; au)l\_j(\; au),\; d\; au$.
The Gramian matrix can be used to test for linear independence of functions. Namely,
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 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
:$x\_1,dots,\; x\_n$
are ''n'' vectors of ''E'', the associated 'Gram matrix' is the symmetric matrix
:$(x\_i|x\_j),$.
The 'Gram determinant' is the determinant of this matrix,
:
(x_2|x_1) & (x_2|x_2) &dots & (x_2|x_n)\
dots&dots&&dots\
(x_n|x_1) & (x_n|x_2) &dots & (x_n|x_n)end{vmatrix}
All eigenvalues of a Gramian matrix are real and non-negative and the matrix is thus also positive semidefinite.
For a general bilinear form ''B'' on a finite-dimensional vector space over any field we can define a Gram matrix ''G'' attached to a basis $x\_1,dots,\; x\_n$ by
:$G\_\{i,j\}\; =\; B(x\_i,x\_j)\; ,$.
The matrix will be symmetric if the bilinear form ''B'' is.
Under change of basis represented by an invertible matrix ''P'', the Gram matrix will change by a matrix congruence to $P^\; op\; G\; P$.

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See also

External links

See also

★ Controllability Gramian

★ Observability Gramian

★ Overlap matrix

External links

★ An application of the Gramian matrix in general linear algebra