Inlinear algebra, theGram matrix (orGramian matrix,Gramian) of vectors${\displaystyle v_1,\dots ,v_n}$ in aninner product space is theHermitian matrix ofinner products, whose entries are given by theinner product${\displaystyle G_{ij}=⟨v_i,v_j⟩}$.^[1] If the vectors${\displaystyle v_1,\dots ,v_n}$ are the columns of matrix${\displaystyle X}$ then the Gram matrix is${\displaystyle X^{†}X}$ in the general case that the vector coordinates are complex numbers, which simplifies to${\displaystyle X^{\mathrm{\top }}X}$ for the case that the vector coordinates are real numbers.

An important application is to computelinear independence: a set of vectors are linearly independent if and only if theGram determinant (thedeterminant of the Gram matrix) is non-zero.

It is named afterJørgen Pedersen Gram.

For finite-dimensional real vectors in${\displaystyle {\mathbb{R}}^n}$ with the usual Euclideandot product, the Gram matrix is${\displaystyle G=V^{\mathrm{\top }}V}$, where${\displaystyle V}$ is a matrix whose columns are the vectors${\displaystyle v_k}$ and${\displaystyle V^{\mathrm{\top }}}$ is itstranspose whose rows are the vectors${\displaystyle v_k^{\mathrm{\top }}}$. Forcomplex vectors in${\displaystyle {\mathbb{C}}^n}$,${\displaystyle G=V^{†}V}$, where${\displaystyle V^{†}}$ is theconjugate transpose of${\displaystyle V}$.

Givensquare-integrable functions${\displaystyle \{{\ell }_i(\cdot ),i=1,\dots ,n\}}$ on the interval${\displaystyle \left[t_0,t_f\right]}$, the Gram matrix${\displaystyle G=\left[G_{ij}\right]}$ is:

${\displaystyle G_{ij}={\int }_{t_0}^{t_f}{\ell }_i^{\ast }(\tau ){\ell }_j(\tau )d\tau .}$

where${\displaystyle {\ell }_i^{\ast }(\tau )}$ is thecomplex conjugate of${\displaystyle {\ell }_i(\tau )}$.

For anybilinear form${\displaystyle B}$ on afinite-dimensionalvector space over anyfield we can define a Gram matrix${\displaystyle G}$ attached to a set of vectors${\displaystyle v_1,\dots ,v_n}$ by${\displaystyle G_{ij}=B\left(v_i,v_j\right)}$. The matrix will be symmetric if the bilinear form${\displaystyle B}$ is symmetric.

• InRiemannian geometry, given an embedded${\displaystyle k}$-dimensionalRiemannian manifold${\displaystyle M\subset {\mathbb{R}}^n}$ and a parametrization${\displaystyle \varphi :U\to M}$ for${\displaystyle (x_1,\dots ,x_k)\in U\subset {\mathbb{R}}^k}$, the volume form${\displaystyle \omega }$ on${\displaystyle M}$ induced by the embedding may be computed using the Gramian of the coordinate tangent vectors:$${\displaystyle \omega =\sqrt{detG}d{x}_1\cdots d{x}_k,G=\left[⟨\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_i},\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_j}⟩\right].}$$ This generalizes the classical surface integral of a parametrized surface${\displaystyle \varphi :U\to S\subset {\mathbb{R}}^3}$ for${\displaystyle (x,y)\in U\subset {\mathbb{R}}^2}$:$${\displaystyle {\int }_{S}fdA={\iint }_{U}f(\varphi (x,y))\left|\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}\times \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}\right|dxdy.}$$
• If the vectors are centeredrandom variables, the Gramian is approximately proportional to thecovariance matrix, with the scaling determined by the number of elements in the vector.
• Inquantum chemistry, the Gram matrix of a set ofbasis vectors is theoverlap matrix.
• Incontrol theory (or more generallysystems theory), thecontrollability Gramian andobservability 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 thefinite 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.
• Inmachine learning,kernel functions are often represented as Gram matrices.^[2] (Also seekernel PCA)
• Since the Gram matrix over the reals is asymmetric matrix, it isdiagonalizable and itseigenvalues are non-negative. The diagonalization of the Gram matrix is thesingular value decomposition.

The Gram matrix issymmetric in the case the inner product is real-valued; it isHermitian in the general, complex case by definition of aninner product.

The Gram matrix ispositive 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:

${\displaystyle x^{†}\mathbf{G}x=\sum _{i,j}{x}_i^{\ast }{x}_j⟨v_i,v_j⟩=\sum _{i,j}⟨x_i{v}_i,x_j{v}_j⟩=⟨\sum _i{x}_i{v}_i,\sum _j{x}_j{v}_j⟩=\Vert \sum _i{x}_i{v}_i{\Vert }^2\ge 0.}$

The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of theinner-product, 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${\displaystyle v_i}$ are linearly independent (that is,$\sum _i{x}_i{v}_i\ne 0$ for all${\displaystyle x}$).^[1]

Given any positive semidefinite matrix${\displaystyle M}$, one can decompose it as:

${\displaystyle M=B^{†}B}$,

where${\displaystyle B^{†}}$ is theconjugate transpose of${\displaystyle B}$ (or${\displaystyle M=B^{\mathsf{\text{T}}}B}$ in the real case).

Here${\displaystyle B}$ is a${\displaystyle k\times n}$ matrix, where${\displaystyle k}$ is therank of${\displaystyle M}$. Various ways to obtain such a decomposition include computing theCholesky decomposition or taking thenon-negative square root of${\displaystyle M}$.

The columns${\displaystyle b^{(1)},\dots ,b^{(n)}}$ of${\displaystyle B}$ can be seen asn vectors in${\displaystyle {\mathbb{C}}^k}$ (ork-dimensional Euclidean space${\displaystyle {\mathbb{R}}^k}$, in the real case). Then

${\displaystyle M_{ij}=b^{(i)}\cdot b^{(j)}}$

where thedot product$a\cdot b=\sum _{\ell =1}^k{a}_{\ell }^{\ast }{b}_{\ell }$ is the usual inner product on${\displaystyle {\mathbb{C}}^k}$.

Thus aHermitian matrix${\displaystyle M}$ is positive semidefinite if and only if it is the Gram matrix of some vectors${\displaystyle b^{(1)},\dots ,b^{(n)}}$. Such vectors are called avector realization of${\displaystyle M}$. The infinite-dimensional analog of this statement isMercer's theorem.

If${\displaystyle M}$ is the Gram matrix of vectors${\displaystyle v_1,\dots ,v_n}$ in${\displaystyle {\mathbb{R}}^k}$ then applying any rotation or reflection of${\displaystyle {\mathbb{R}}^k}$ (anyorthogonal transformation, that is, anyEuclidean isometry preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any${\displaystyle k\times k}$orthogonal matrix${\displaystyle Q}$, the Gram matrix of${\displaystyle Q{v}_1,\dots ,Q{v}_n}$ is also${\displaystyle M}$.

This is the only way in which two real vector realizations of${\displaystyle M}$ can differ: the vectors${\displaystyle v_1,\dots ,v_n}$ are unique up toorthogonal transformations. In other words, the dot products${\displaystyle v_i\cdot v_j}$ and${\displaystyle w_i\cdot w_j}$ are equal if and only if some rigid transformation of${\displaystyle {\mathbb{R}}^k}$ transforms the vectors${\displaystyle v_1,\dots ,v_n}$ to${\displaystyle w_1,\dots ,w_n}$ and 0 to 0.

The same holds in the complex case, withunitary transformations in place of orthogonal ones.That is, if the Gram matrix of vectors${\displaystyle v_1,\dots ,v_n}$ is equal to the Gram matrix of vectors${\displaystyle w_1,\dots ,w_n}$ in${\displaystyle {\mathbb{C}}^k}$ then there is aunitary${\displaystyle k\times k}$ matrix${\displaystyle U}$ (meaning${\displaystyle U^{†}U=I}$) such that${\displaystyle v_i=U{w}_i}$ for${\displaystyle i=1,\dots ,n}$.^[3]

• Because${\displaystyle G=G^{†}}$, it is necessarily the case that${\displaystyle G}$ and${\displaystyle G^{†}}$ commute. That is, a real or complex Gram matrix${\displaystyle G}$ is also anormal matrix.
• The Gram matrix of anyorthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a realrotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of aunitary matrix is the identity matrix.
• The rank of the Gram matrix of vectors in${\displaystyle {\mathbb{R}}^k}$ or${\displaystyle {\mathbb{C}}^k}$ equals the dimension of the spacespanned by these vectors.^[1]

TheGram determinant orGramian is the determinant of the Gram matrix:

If${\displaystyle v_1,\dots ,v_n}$ are vectors in${\displaystyle {\mathbb{R}}^m}$ then it is the square of then-dimensional volume of theparallelotope formed by the vectors. In particular, the vectors arelinearly independentif and only if the parallelotope has nonzeron-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix isnonsingular. Whenn >m the determinant and volume are zero. Whenn =m, this reduces to the standard theorem that the absolute value of the determinant ofnn-dimensional vectors is then-dimensional volume. The volume of thesimplex formed by the vectors isVolume(parallelotope) /n!.

When${\displaystyle v_1,\dots ,v_n}$ are linearly independent, the distance between a point${\displaystyle x}$ and the linear span of${\displaystyle v_1,\dots ,v_n}$ is${\displaystyle \sqrt{\frac{|G(x,v_1,\dots ,v_n)|}{|G(v_1,\dots ,v_n)|}}}$.

Consider the moment problem: given${\displaystyle c_1,\dots ,c_n\in \mathbb{C}}$, find a vector${\displaystyle v}$ such that$⟨v,v_i⟩=c_i$, for all$1⩽i⩽n$. There exists a unique solution with minimal norm:^[4]: 38The Gram determinant can also be expressed in terms of theexterior product of vectors by

${\displaystyle |G(v_1,\dots ,v_n)|=\Vert v_1\wedge \cdots \wedge v_n{\Vert }^2.}$

The Gram determinant therefore supplies aninner product for the space⁠${\displaystyle {\bigwedge }^n(V)}$⁠. If anorthonormal basise_i,i = 1, 2, ...,n on⁠${\displaystyle V}$⁠ is given, the vectors

will constitute an orthonormal basis ofn-dimensional volumes on the space⁠${\displaystyle {\bigwedge }^n(V)}$⁠. Then the Gram determinant${\displaystyle |G(v_1,\dots ,v_n)|}$ amounts to ann-dimensionalPythagorean Theorem for the volume of the parallelotope formed by the vectors${\displaystyle v_1\wedge \cdots \wedge v_n}$ in terms of its projections onto the basis volumes${\displaystyle e_{i_1}\wedge \cdots \wedge e_{i_n}}$.

When the vectors${\displaystyle v_1,\dots ,v_n\in {\mathbb{R}}^m}$ are defined from the positions of points${\displaystyle p_1,\dots ,p_n}$ relative to some reference point${\displaystyle p_{n+1}}$,

$${\displaystyle (v_1,v_2,\dots ,v_n)=(p_1-p_{n+1},p_2-p_{n+1},\dots ,p_n-p_{n+1}),}$$

then the Gram determinant can be written as the difference of two Gram determinants,

$${\displaystyle |G(v_1,\dots ,v_n)|=|G((p_1,1),\dots ,(p_{n+1},1))|-|G(p_1,\dots ,p_{n+1})|,}$$

where each${\displaystyle (p_j,1)}$ is the corresponding point${\displaystyle p_j}$ supplemented with the coordinate value of 1 for an${\displaystyle (m+1)}$-st dimension.^{[citation needed]} Note that in the common case thatn =m, the second term on the right-hand side will be zero.

Given a set of linearly independent vectors${\displaystyle \{v_i\}}$ with Gram matrix${\displaystyle G}$ defined by${\displaystyle G_{ij}:=⟨v_i,v_j⟩}$, one can construct an orthonormal basis

${\displaystyle u_i:=\sum _j(G^{-1/2}{)}_{ji}{v}_j.}$

In matrix notation,${\displaystyle U=V{G}^{-1/2}}$, where${\displaystyle U}$ has orthonormal basis vectors${\displaystyle \{u_i\}}$ and the matrix${\displaystyle V}$ is composed of the given column vectors${\displaystyle \{v_i\}}$.

The matrix${\displaystyle G^{-1/2}}$ is guaranteed to exist. Indeed,${\displaystyle G}$ is Hermitian, and so can be decomposed as${\displaystyle G=UD{U}^{†}}$ with${\displaystyle U}$ a unitary matrix and${\displaystyle D}$ a real diagonal matrix. Additionally, the${\displaystyle v_i}$ are linearly independent if and only if${\displaystyle G}$ is positive definite, which implies that the diagonal entries of${\displaystyle D}$ are positive.${\displaystyle G^{-1/2}}$ is therefore uniquely defined by${\displaystyle G^{-1/2}:=U{D}^{-1/2}{U}^{†}}$. One can check that these new vectors are orthonormal:

where we used${\displaystyle (G^{-1/2}{)}^{†}=G^{-1/2}}$.

• Controllability Gramian
• Observability Gramian

• Horn, Roger A.; Johnson, Charles R. (2013).Matrix Analysis (2nd ed.).Cambridge University Press.ISBN 978-0-521-54823-6.

• "Gram matrix",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Volumes of parallelograms by Frank Jones

• Systems theory
• Matrices (mathematics)
• Determinants
• Analytic geometry
• Kernel methods for machine learning

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