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Gram matrix
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<h2 id="examples">Examples</h2> <p>For finite-dimensional real vectors in R n {\displaystyle \mathbb {R} ^{n}} with the usual Euclidean <a href="/facts/Dot_product/tNz8MLjT">dot product</a>, the Gram matrix is G = V ⊤ V {\displaystyle G=V^{\top }V} , where V {\displaystyle V} is a matrix whose columns are the vectors v k {\displaystyle v_{k}} and V ⊤ {\displaystyle V^{\top }} is its <a href="/facts/Transpose/8wmsagGS">transpose</a> whose rows are the vectors v k ⊤ {\displaystyle v_{k}^{\top }} . For <a href="/facts/Complex_number/2w7mqI7e">complex</a> vectors in C n {\displaystyle \mathbb {C} ^{n}} , G = V † V {\displaystyle G=V^{\dagger }V} , where V † {\displaystyle V^{\dagger }} is the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> of V {\displaystyle V} . </p><p>Given <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable functions</a> { ℓ i ( ⋅ ) , i = 1 , … , n } {\displaystyle \{\ell _{i}(\cdot ),\,i=1,\dots ,n\}} on the interval [ t 0 , t f ] {\displaystyle \left[t_{0},t_{f}\right]} , the Gram matrix G = [ G i j ] {\displaystyle G=\left[G_{ij}\right]} is: </p> G i j = ∫ t 0 t f ℓ i ∗ ( τ ) ℓ j ( τ ) d τ . {\displaystyle G_{ij}=\int _{t_{0}}^{t_{f}}\ell _{i}^{*}(\tau )\ell _{j}(\tau )\,d\tau .} <p>where ℓ i ∗ ( τ ) {\displaystyle \ell _{i}^{*}(\tau )} is the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of ℓ i ( τ ) {\displaystyle \ell _{i}(\tau )} . </p><p>For any <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a> B {\displaystyle B} on a <a href="/facts/Finite-dimensional/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> we can define a Gram matrix G {\displaystyle G} attached to a set of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} by G i j = B ( v i , v j ) {\displaystyle G_{ij}=B\left(v_{i},v_{j}\right)} . The matrix will be symmetric if the bilinear form B {\displaystyle B} is symmetric. </p> <h3>Applications</h3> <ul><li>In <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>, given an embedded k {\displaystyle k} -dimensional <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> M ⊂ R n {\displaystyle M\subset \mathbb {R} ^{n}} and a parametrization ϕ : U → M {\displaystyle \phi :U\to M} for ( x 1 , … , x k ) ∈ U ⊂ R k {\displaystyle (x_{1},\ldots ,x_{k})\in U\subset \mathbb {R} ^{k}} , the volume form ω {\displaystyle \omega } on M {\displaystyle M} induced by the embedding may be computed using the Gramian of the coordinate tangent vectors: ω = det G d x 1 ⋯ d x k , G = [ ⟨ ∂ ϕ ∂ x i , ∂ ϕ ∂ x j ⟩ ] . {\displaystyle \omega ={\sqrt {\det G}}\ dx_{1}\cdots dx_{k},\quad G=\left[\left\langle {\frac {\partial \phi }{\partial x_{i}}},{\frac {\partial \phi }{\partial x_{j}}}\right\rangle \right].} This generalizes the classical surface integral of a parametrized surface ϕ : U → S ⊂ R 3 {\displaystyle \phi :U\to S\subset \mathbb {R} ^{3}} for ( x , y ) ∈ U ⊂ R 2 {\displaystyle (x,y)\in U\subset \mathbb {R} ^{2}} : ∫ S f d A = ∬ U f ( ϕ ( x , y ) ) | ∂ ϕ ∂ x × ∂ ϕ ∂ y | d x d y . {\displaystyle \int _{S}f\ dA=\iint _{U}f(\phi (x,y))\,\left|{\frac {\partial \phi }{\partial x}}\,{\times }\,{\frac {\partial \phi }{\partial y}}\right|\,dx\,dy.} </li> <li>If the vectors are centered <a href="/facts/Random_variable/TwTBXnLT">random variables</a>, the Gramian is approximately proportional to the <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a>, with the scaling determined by the number of elements in the vector.</li> <li>In <a href="/facts/Quantum_chemistry/H9AXV0lx">quantum chemistry</a>, the Gram matrix of a set of <a href="/facts/Basis_vectors/89IPoN6c">basis vectors</a> is the <a href="/facts/Overlap_matrix/IECydUNY">overlap matrix</a>.</li> <li>In <a href="/facts/Control_theory/aIuERpn6">control theory</a> (or more generally <a href="/facts/Systems_theory/C56Ck7DQ">systems theory</a>), the <a href="/facts/Controllability_Gramian/gcLF3Bm1">controllability Gramian</a> and <a href="/facts/Observability_Gramian/ghFmHQQ9">observability Gramian</a> determine properties of a linear system.</li> <li>Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp. 79–94).</li> <li>In the <a href="/facts/Finite_element_method/eUcfP1vn">finite element method</a>, 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.</li> <li>In <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a>, <a href="/facts/Kernel_function/Ws3zkxfl">kernel functions</a> are often represented as Gram matrices.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> (Also see <a href="/facts/Kernel_principal_component_analysis/VLNWSISo">kernel PCA</a>)</li> <li>Since the Gram matrix over the reals is a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a>, it is <a href="/facts/Diagonalizable/y3MkyTCy">diagonalizable</a> and its <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> are non-negative. The diagonalization of the Gram matrix is the <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a>.</li></ul> <h2 id="properties">Properties</h2> <h3>Positive-semidefiniteness</h3> <p>The Gram matrix is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> in the case the inner product is real-valued; it is <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian</a> in the general, complex case by definition of an <a href="/facts/Inner_product/HoyjElEy">inner product</a>. </p><p>The Gram matrix is <a href="/facts/Positive-semidefinite_matrix/Hbr6AuPS">positive semidefinite</a>, 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: </p> x † G x = ∑ i , j x i ∗ x j ⟨ v i , v j ⟩ = ∑ i , j ⟨ x i v i , x j v j ⟩ = ⟨ ∑ i x i v i , ∑ j x j v j ⟩ = ‖ ∑ i x i v i ‖ 2 ≥ 0. {\displaystyle x^{\dagger }\mathbf {G} x=\sum _{i,j}x_{i}^{*}x_{j}\left\langle v_{i},v_{j}\right\rangle =\sum _{i,j}\left\langle x_{i}v_{i},x_{j}v_{j}\right\rangle ={\biggl \langle }\sum _{i}x_{i}v_{i},\sum _{j}x_{j}v_{j}{\biggr \rangle }={\biggl \|}\sum _{i}x_{i}v_{i}{\biggr \|}^{2}\geq 0.} <p>The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of the <a href="/facts/Inner-product/HoyjElEy">inner-product</a>, and the last from the positive definiteness of the inner product. Note that this also shows that the Gramian matrix is positive definite if and only if the vectors v i {\displaystyle v_{i}} are linearly independent (that is, ∑ i x i v i ≠ 0 {\textstyle \sum _{i}x_{i}v_{i}\neq 0} for all x {\displaystyle x} ).<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h3>Finding a vector realization</h3> <p class="note">See also: <a href="/facts/Positive_definite_matrix/Hbr6AuPS">Positive definite matrix § Decomposition</a></p> <p>Given any positive semidefinite matrix M {\displaystyle M} , one can decompose it as: </p> M = B † B {\displaystyle M=B^{\dagger }B} , <p>where B † {\displaystyle B^{\dagger }} is the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> of B {\displaystyle B} (or M = B T B {\displaystyle M=B^{\textsf {T}}B} in the real case). </p><p>Here B {\displaystyle B} is a k × n {\displaystyle k\times n} matrix, where k {\displaystyle k} is the <a href="/facts/Matrix_rank/Yp9b5LK3">rank</a> of M {\displaystyle M} . Various ways to obtain such a decomposition include computing the <a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky decomposition</a> or taking the <a href="/facts/Square_root_of_a_matrix/uWGkQQqN">non-negative square root</a> of M {\displaystyle M} . </p><p>The columns b ( 1 ) , … , b ( n ) {\displaystyle b^{(1)},\dots ,b^{(n)}} of B {\displaystyle B} can be seen as <i>n</i> vectors in C k {\displaystyle \mathbb {C} ^{k}} (or <i>k</i>-dimensional Euclidean space R k {\displaystyle \mathbb {R} ^{k}} , in the real case). Then </p> M i j = b ( i ) ⋅ b ( j ) {\displaystyle M_{ij}=b^{(i)}\cdot b^{(j)}} <p>where the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> a ⋅ b = ∑ ℓ = 1 k a ℓ ∗ b ℓ {\textstyle a\cdot b=\sum _{\ell =1}^{k}a_{\ell }^{*}b_{\ell }} is the usual inner product on C k {\displaystyle \mathbb {C} ^{k}} . </p><p>Thus a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> M {\displaystyle M} is positive semidefinite if and only if it is the Gram matrix of some vectors b ( 1 ) , … , b ( n ) {\displaystyle b^{(1)},\dots ,b^{(n)}} . Such vectors are called a vector realization of M {\displaystyle M} . The infinite-dimensional analog of this statement is <a href="/facts/Mercer%27s_theorem/6CmBHnHc">Mercer's theorem</a>. </p> <h3>Uniqueness of vector realizations</h3> <p>If M {\displaystyle M} is the Gram matrix of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} in R k {\displaystyle \mathbb {R} ^{k}} then applying any rotation or reflection of R k {\displaystyle \mathbb {R} ^{k}} (any <a href="/facts/Orthogonal_transformation/s7LGBDFX">orthogonal transformation</a>, that is, any <a href="/facts/Euclidean_isometry/0KzWYQXl">Euclidean isometry</a> preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any k × k {\displaystyle k\times k} <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a> Q {\displaystyle Q} , the Gram matrix of Q v 1 , … , Q v n {\displaystyle Qv_{1},\dots ,Qv_{n}} is also M {\displaystyle M} . </p><p>This is the only way in which two real vector realizations of M {\displaystyle M} can differ: the vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} are unique up to <a href="/facts/Orthogonal_transformation/s7LGBDFX">orthogonal transformations</a>. In other words, the dot products v i ⋅ v j {\displaystyle v_{i}\cdot v_{j}} and w i ⋅ w j {\displaystyle w_{i}\cdot w_{j}} are equal if and only if some rigid transformation of R k {\displaystyle \mathbb {R} ^{k}} transforms the vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} to w 1 , … , w n {\displaystyle w_{1},\dots ,w_{n}} and 0 to 0. </p><p>The same holds in the complex case, with <a href="/facts/Unitary_transformation/DyvU0nms">unitary transformations</a> in place of orthogonal ones. That is, if the Gram matrix of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} is equal to the Gram matrix of vectors w 1 , … , w n {\displaystyle w_{1},\dots ,w_{n}} in C k {\displaystyle \mathbb {C} ^{k}} then there is a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary</a> k × k {\displaystyle k\times k} matrix U {\displaystyle U} (meaning U † U = I {\displaystyle U^{\dagger }U=I} ) such that v i = U w i {\displaystyle v_{i}=Uw_{i}} for i = 1 , … , n {\displaystyle i=1,\dots ,n} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h3>Other properties</h3> <ul><li>Because G = G † {\displaystyle G=G^{\dagger }} , it is necessarily the case that G {\displaystyle G} and G † {\displaystyle G^{\dagger }} commute. That is, a real or complex Gram matrix G {\displaystyle G} is also a <a href="/facts/Normal_matrix/WVKxrEK9">normal matrix</a>.</li> <li>The Gram matrix of any <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a real <a href="/facts/Rotation_matrix/c2K2Ftl8">rotation matrix</a> is the identity matrix. Likewise, the Gram matrix of the rows or columns of a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a> is the identity matrix.</li> <li>The rank of the Gram matrix of vectors in R k {\displaystyle \mathbb {R} ^{k}} or C k {\displaystyle \mathbb {C} ^{k}} equals the dimension of the space <a href="/facts/Linear_span/vPjclEtb">spanned</a> by these vectors.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <h2 id="gram-determinant">Gram determinant</h2> <p>The Gram determinant or Gramian is the determinant of the Gram matrix: | G ( v 1 , … , v n ) | = | ⟨ v 1 , v 1 ⟩ ⟨ v 1 , v 2 ⟩ … ⟨ v 1 , v n ⟩ ⟨ v 2 , v 1 ⟩ ⟨ v 2 , v 2 ⟩ … ⟨ v 2 , v n ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v n , v 1 ⟩ ⟨ v n , v 2 ⟩ … ⟨ v n , v n ⟩ | . {\displaystyle {\bigl |}G(v_{1},\dots ,v_{n}){\bigr |}={\begin{vmatrix}\langle v_{1},v_{1}\rangle &\langle v_{1},v_{2}\rangle &\dots &\langle v_{1},v_{n}\rangle \\\langle v_{2},v_{1}\rangle &\langle v_{2},v_{2}\rangle &\dots &\langle v_{2},v_{n}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle v_{n},v_{1}\rangle &\langle v_{n},v_{2}\rangle &\dots &\langle v_{n},v_{n}\rangle \end{vmatrix}}.} </p><p>If v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} are vectors in R m {\displaystyle \mathbb {R} ^{m}} then it is the square of the <i>n</i>-dimensional volume of the <a href="/facts/Parallelepiped/Qn7cDgSr">parallelotope</a> formed by the vectors. In particular, the vectors are <a href="/facts/Linear_independence/8JrHfIIa">linearly independent</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the parallelotope has nonzero <i>n</i>-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is <a href="/facts/Non-singular_matrix/fPqXk3V8">nonsingular</a>. When <i>n</i> > <i>m</i> the determinant and volume are zero. When <i>n</i> = <i>m</i>, this reduces to the standard theorem that the absolute value of the determinant of <i>n</i> <i>n</i>-dimensional vectors is the <i>n</i>-dimensional volume. The Gram determinant is also useful for computing the volume of the <a href="/facts/Simplex/0FCfNRN8">simplex</a> formed by the vectors; its volume is Volume(parallelotope) / <i>n</i>!. </p><p>The Gram determinant can also be expressed in terms of the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a> of vectors by </p> | G ( v 1 , … , v n ) | = ‖ v 1 ∧ ⋯ ∧ v n ‖ 2 . {\displaystyle {\bigl |}G(v_{1},\dots ,v_{n}){\bigr |}=\|v_{1}\wedge \cdots \wedge v_{n}\|^{2}.} <p>The Gram determinant therefore supplies an <a href="/facts/Exterior_product/rdN4TvpR">inner product</a> for the space ⋀ n ( V ) {\displaystyle {\textstyle \bigwedge }^{\!n}(V)} . If an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> <i>e</i><i>i</i>, <i>i</i> = 1, 2, ..., <i>n</i> on V {\displaystyle V} is given, the vectors </p> e i 1 ∧ ⋯ ∧ e i n , i 1 < ⋯ < i n , {\displaystyle e_{i_{1}}\wedge \cdots \wedge e_{i_{n}},\quad i_{1}<\cdots <i_{n},} <p>will constitute an orthonormal basis of <i>n</i>-dimensional volumes on the space ⋀ n ( V ) {\displaystyle {\textstyle \bigwedge }^{\!n}(V)} . Then the Gram determinant | G ( v 1 , … , v n ) | {\displaystyle {\bigl |}G(v_{1},\dots ,v_{n}){\bigr |}} amounts to an <i>n</i>-dimensional <a href="/facts/Pythagorean_Theorem/Lh6x2Kr5">Pythagorean Theorem</a> for the volume of the parallelotope formed by the vectors v 1 ∧ ⋯ ∧ v n {\displaystyle v_{1}\wedge \cdots \wedge v_{n}} in terms of its projections onto the basis volumes e i 1 ∧ ⋯ ∧ e i n {\displaystyle e_{i_{1}}\wedge \cdots \wedge e_{i_{n}}} . </p><p>When the vectors v 1 , … , v n ∈ R m {\displaystyle v_{1},\ldots ,v_{n}\in \mathbb {R} ^{m}} are defined from the positions of points p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} relative to some reference point p n + 1 {\displaystyle p_{n+1}} , </p> ( v 1 , v 2 , … , v n ) = ( p 1 − p n + 1 , p 2 − p n + 1 , … , p n − p n + 1 ) , {\displaystyle (v_{1},v_{2},\ldots ,v_{n})=(p_{1}-p_{n+1},p_{2}-p_{n+1},\ldots ,p_{n}-p_{n+1})\,,} <p>then the Gram determinant can be written as the difference of two Gram determinants, </p> | G ( v 1 , … , v n ) | = | G ( ( p 1 , 1 ) , … , ( p n + 1 , 1 ) ) | − | G ( p 1 , … , p n + 1 ) | , {\displaystyle {\bigl |}G(v_{1},\dots ,v_{n}){\bigr |}={\bigl |}G((p_{1},1),\dots ,(p_{n+1},1)){\bigr |}-{\bigl |}G(p_{1},\dots ,p_{n+1}){\bigr |}\,,} <p>where each ( p j , 1 ) {\displaystyle (p_{j},1)} is the corresponding point p j {\displaystyle p_{j}} supplemented with the coordinate value of 1 for an ( m + 1 ) {\displaystyle (m+1)} -st dimension. Note that in the common case that <i>n</i> = <i>m</i>, the second term on the right-hand side will be zero. </p> <h2 id="constructing-an-orthonormal-basis">Constructing an orthonormal basis</h2> <p>Given a set of linearly independent vectors { v i } {\displaystyle \{v_{i}\}} with Gram matrix G {\displaystyle G} defined by G i j := ⟨ v i , v j ⟩ {\displaystyle G_{ij}:=\langle v_{i},v_{j}\rangle } , one can construct an orthonormal basis </p> u i := ∑ j ( G − 1 / 2 ) j i v j . {\displaystyle u_{i}:=\sum _{j}{\bigl (}G^{-1/2}{\bigr )}_{ji}v_{j}.} <p>In matrix notation, U = V G − 1 / 2 {\displaystyle U=VG^{-1/2}} , where U {\displaystyle U} has orthonormal basis vectors { u i } {\displaystyle \{u_{i}\}} and the matrix V {\displaystyle V} is composed of the given column vectors { v i } {\displaystyle \{v_{i}\}} . </p><p>The matrix G − 1 / 2 {\displaystyle G^{-1/2}} is guaranteed to exist. Indeed, G {\displaystyle G} is Hermitian, and so can be decomposed as G = U D U † {\displaystyle G=UDU^{\dagger }} with U {\displaystyle U} a unitary matrix and D {\displaystyle D} a real diagonal matrix. Additionally, the v i {\displaystyle v_{i}} are linearly independent if and only if G {\displaystyle G} is positive definite, which implies that the diagonal entries of D {\displaystyle D} are positive. G − 1 / 2 {\displaystyle G^{-1/2}} is therefore uniquely defined by G − 1 / 2 := U D − 1 / 2 U † {\displaystyle G^{-1/2}:=UD^{-1/2}U^{\dagger }} . One can check that these new vectors are orthonormal: </p> ⟨ u i , u j ⟩ = ∑ i ′ ∑ j ′ ⟨ ( G − 1 / 2 ) i ′ i v i ′ , ( G − 1 / 2 ) j ′ j v j ′ ⟩ = ∑ i ′ ∑ j ′ ( G − 1 / 2 ) i i ′ G i ′ j ′ ( G − 1 / 2 ) j ′ j = ( G − 1 / 2 G G − 1 / 2 ) i j = δ i j {\displaystyle {\begin{aligned}\langle u_{i},u_{j}\rangle &=\sum _{i'}\sum _{j'}{\Bigl \langle }{\bigl (}G^{-1/2}{\bigr )}_{i'i}v_{i'},{\bigl (}G^{-1/2}{\bigr )}_{j'j}v_{j'}{\Bigr \rangle }\\[10mu]&=\sum _{i'}\sum _{j'}{\bigl (}G^{-1/2}{\bigr )}_{ii'}G_{i'j'}{\bigl (}G^{-1/2}{\bigr )}_{j'j}\\[8mu]&={\bigl (}G^{-1/2}GG^{-1/2}{\bigr )}_{ij}=\delta _{ij}\end{aligned}}} <p>where we used ( G − 1 / 2 ) † = G − 1 / 2 {\displaystyle {\bigl (}G^{-1/2}{\bigr )}^{\dagger }=G^{-1/2}} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Controllability_Gramian/gcLF3Bm1">Controllability Gramian</a></li> <li><a href="/facts/Observability_Gramian/ghFmHQQ9">Observability Gramian</a></li></ul> <ul><li>Horn, Roger A.; Johnson, Charles R. (2013). <i>Matrix Analysis</i> (2nd ed.). <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-54823-6.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Gram_matrix">"Gram matrix"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><i><a href="https://mathweb.rice.edu/honors-calculus">Volumes of parallelograms</a></i> by Frank Jones</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Horn & Johnson 2013, p. 441, p.441, Theorem 7.2.10 - Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lanckriet, G. R. G.; Cristianini, N.; Bartlett, P.; Ghaoui, L. E.; Jordan, M. I. (2004). "Learning the kernel matrix with semidefinite programming". Journal of Machine Learning Research. 5: 27–72 [p. 29]. <a href="https://dl.acm.org/citation.cfm?id=894170" target="_blank">https://dl.acm.org/citation.cfm?id=894170</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Horn & Johnson 2013, p. 441, p.441, Theorem 7.2.10 - Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Horn & Johnson (2013), p. 452, Theorem 7.3.11 - Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Horn & Johnson 2013, p. 441, p.441, Theorem 7.2.10 - Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>