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| [[File:Gram–Schmidt process.svg|right|frame|The first two steps of the Gram–Schmidt process]]
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| In [[mathematics]], particularly [[linear algebra]] and [[numerical analysis]], the '''Gram–Schmidt process''' is a method for [[Orthonormal basis|orthonormalising]] a set of [[vector (geometry)|vectors]] in an [[inner product space]], most commonly the [[Euclidean space]] '''R'''<sup>''n''</sup>. The Gram–Schmidt process takes a [[finite set|finite]], [[linearly independent]] set ''S'' = {''v''<sub>1</sub>, …, ''v''<sub>''k''</sub>} for {{nowrap|''k'' ≤ ''n''}} and generates an [[orthogonal set]] {{nowrap|''S′'' {{=}} {''u''<sub>1</sub>, …, ''u''<sub>''k''</sub>} }} that spans the same ''k''-dimensional subspace of '''R'''<sup>''n''</sup> as ''S''.
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| The method is named after [[Jørgen Pedersen Gram]] and [[Erhard Schmidt]] but it appeared earlier in the work of [[Laplace]] and [[Cauchy]]. In the theory of [[Lie group decompositions]] it is generalized by the [[Iwasawa decomposition]].<ref>Cheney, Ward; Kincaid, David: ''Linear Algebra: Theory and Applications''. Sudbury, Ma: 2009.Pg. 544, 558.</ref>
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| The application of the Gram–Schmidt process to the column vectors of a full column [[rank (linear algebra)|rank]] [[matrix (mathematics)|matrix]] yields the [[QR decomposition]] (it is decomposed into an [[orthogonal matrix|orthogonal]] and a [[triangular matrix]]).
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| == The Gram–Schmidt process ==
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| [[File:Gram-Schmidt orthonormalization process.gif|frame|right|The Gram-Schmidt process being executed on three linearly independent, non-orthogonal vectors of a basis for '''R'''<sup>3</sup>. Click on image for details.]]
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| We define the [[projection (linear algebra)|projection]] [[operator (mathematics)|operator]] by
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| :<math>\mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{u}, \mathbf{v}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} , </math>
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| where <math>\langle \mathbf{u}, \mathbf{v}\rangle</math> denotes the [[inner product]] of the vectors '''u''' and '''v'''. This operator projects the vector '''v''' orthogonally onto the line spanned by vector '''u'''. If '''u'''=0, we define <math>\mathrm{proj}_0\,(\mathbf{v}) := 0</math>. i.e., the projection map <math>\mathrm{proj}_0</math> is the zero map, sending every vector to the zero vector.
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| The Gram–Schmidt process then works as follows:
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| : <math>
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| \begin{align}
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| \mathbf{u}_1 & = \mathbf{v}_1, & \mathbf{e}_1 & = {\mathbf{u}_1 \over \|\mathbf{u}_1\|} \\
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| \mathbf{u}_2 & = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2),
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| & \mathbf{e}_2 & = {\mathbf{u}_2 \over \|\mathbf{u}_2\|} \\
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| \mathbf{u}_3 & = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3), & \mathbf{e}_3 & = {\mathbf{u}_3 \over \|\mathbf{u}_3\|} \\
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| \mathbf{u}_4 & = \mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_3}\,(\mathbf{v}_4), & \mathbf{e}_4 & = {\mathbf{u}_4 \over \|\mathbf{u}_4\|} \\
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| & {}\ \ \vdots & & {}\ \ \vdots \\
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| \mathbf{u}_k & = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k), & \mathbf{e}_k & = {\mathbf{u}_k\over \|\mathbf{u}_k \|}.
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| \end{align}
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| </math>
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| The sequence '''u'''<sub>1</sub>, ..., '''u'''<sub>''k''</sub> is the required system of orthogonal vectors, and the normalized vectors '''e'''<sub>1</sub>, ..., '''e'''<sub>''k''</sub> form an [[orthonormal|ortho''normal'']] set. The calculation of the sequence '''u'''<sub>1</sub>, ..., '''u'''<sub>''k''</sub> is known as ''Gram–Schmidt [[orthogonalization]]'', while the calculation of the sequence '''e'''<sub>1</sub>, ..., '''e'''<sub>''k''</sub> is known as ''Gram–Schmidt [[orthonormalization]]'' as the vectors are normalized.
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| To check that these formulas yield an orthogonal sequence, first compute ‹ '''u'''<sub>1</sub>,'''u'''<sub>2</sub> › by substituting the above formula for '''u'''<sub>2</sub>: we get zero. Then use this to compute ‹ '''u'''<sub>1</sub>,'''u'''<sub>3</sub> › again by substituting the formula for '''u'''<sub>3</sub>: we get zero. The general proof proceeds by [[mathematical induction]].
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| Geometrically, this method proceeds as follows: to compute '''u'''<sub>''i''</sub>, it projects '''v'''<sub>''i''</sub> orthogonally onto the subspace ''U'' generated by '''u'''<sub>1</sub>, ..., '''u'''<sub>''i''−1</sub>, which is the same as the subspace generated by '''v'''<sub>1</sub>, ..., '''v'''<sub>''i''−1</sub>. The vector '''u'''<sub>''i''</sub> is then defined to be the difference between '''v'''<sub>''i''</sub> and this projection, guaranteed to be orthogonal to all of the vectors in the subspace ''U''.
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| The Gram–Schmidt process also applies to a linearly independent [[countably infinite]] sequence {'''v'''<sub>''i''</sub>}<sub>''i''</sub>. The result is an orthogonal (or orthonormal) sequence {'''u'''<sub>''i''</sub>}<sub>''i''</sub> such that for natural number ''n'':
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| the algebraic span of '''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub> is the same as that of '''u'''<sub>1</sub>, ..., '''u'''<sub>''n''</sub>.
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| If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the '''0''' vector on the ''i''th step, assuming that '''v'''<sub>''i''</sub> is a linear combination of {{nowrap|'''v'''<sub>1</sub>, ..., '''v'''<sub>''i''−1</sub>}}. If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs.
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| A variant of the Gram–Schmidt process using [[transfinite recursion]] applied to a (possibly uncountably) infinite sequence of vectors <math>(v_\alpha)_{\alpha<\lambda}</math> yields a set of orthonormal vectors <math>(u_\alpha)_{\alpha<\kappa}</math> with <math>\kappa\leq\lambda</math> such that for any <math>\alpha\leq\lambda</math>, the [[Complete_space#Completion|completion]] of the span of <math>\lbrace u_\beta : \beta<\min(\alpha,\kappa)\rbrace</math> is the same as that of <math>\lbrace v_\beta:\beta<\alpha\rbrace</math>. In particular, when applied to a (algebraic) basis of a [[Hilbert space]] (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis. Note that in the general case often the strict inequality <math>\kappa<\lambda</math> holds, even if the starting set was linearly independent, and the span of <math>(u_\alpha)_{\alpha<\kappa}</math> need not be a subspace of the span of <math>(v_\alpha)_{\alpha<\lambda}</math> (rather, it's a subspace of its completion).
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| == Example ==
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| Consider the following set of vectors in '''R'''<sup>2</sup> (with the conventional inner product)
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| :<math>S = \left\lbrace\mathbf{v}_1=\begin{pmatrix} 3 \\ 1\end{pmatrix}, \mathbf{v}_2=\begin{pmatrix}2 \\2\end{pmatrix}\right\rbrace.</math>
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| Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors:
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| :<math>\mathbf{u}_1=\mathbf{v}_1=\begin{pmatrix}3\\1\end{pmatrix}</math>
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| :<math> \mathbf{u}_2 = \mathbf{v}_2 - \mathrm{proj}_{\mathbf{u}_1} \, (\mathbf{v}_2) = \begin{pmatrix}2\\2\end{pmatrix} - \mathrm{proj}_{({3 \atop 1})} \, ({\begin{pmatrix}2\\2\end{pmatrix})} = \begin{pmatrix} -2/5 \\6/5 \end{pmatrix}. </math>
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| We check that the vectors '''u'''<sub>1</sub> and '''u'''<sub>2</sub> are indeed orthogonal:
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| :<math>\langle\mathbf{u}_1,\mathbf{u}_2\rangle = \left\langle \begin{pmatrix}3\\1\end{pmatrix}, \begin{pmatrix}-2/5\\6/5\end{pmatrix} \right\rangle = -\frac65 + \frac65 = 0,</math>
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| noting that if the dot product of two vectors is ''0'' then they are orthogonal.
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| We can then normalize the vectors by dividing out their sizes as shown above:
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| :<math>\mathbf{e}_1 = {1 \over \sqrt {10}}\begin{pmatrix}3\\1\end{pmatrix}</math>
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| :<math>\mathbf{e}_2 = {1 \over \sqrt{40 \over 25}} \begin{pmatrix}-2/5\\6/5\end{pmatrix}
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| = {1\over\sqrt{10}} \begin{pmatrix}-1\\3\end{pmatrix}. </math>
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| == Numerical stability ==
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| When this process is implemented on a computer, the vectors <math>\mathbf{u}_k</math> are often not quite orthogonal, due to [[round-off error|rounding errors]]. For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt") this loss of orthogonality is particularly bad; therefore, it is said that the (classical) Gram–Schmidt process is [[numerical stability|numerically unstable]].
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| The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as '''modified Gram-Schmidt''' or MGS.
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| This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic.
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| Instead of computing the vector '''u'''<sub>''k''</sub> as
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| :<math> \mathbf{u}_k = \mathbf{v}_k - \mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_k) - \mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_k) - \cdots - \mathrm{proj}_{\mathbf{u}_{k-1}}\,(\mathbf{v}_k), </math> | |
| it is computed as {{clarify|Are the Uk in the formula as calculated above, or the result of the recursive calculation?|date=April 2012}}
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| :<math> \begin{align}
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| \mathbf{u}_k^{(1)} &= \mathbf{v}_k - \mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_k), \\
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| \mathbf{u}_k^{(2)} &= \mathbf{u}_k^{(1)} - \mathrm{proj}_{\mathbf{u}_2} \, (\mathbf{u}_k^{(1)}), \\
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| & \,\,\, \vdots \\
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| \mathbf{u}_k^{(k-2)} &= \mathbf{u}_k^{(k-3)} - \mathrm{proj}_{\mathbf{u}_{k-2}} \, (\mathbf{u}_k^{(k-3)}), \\
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| \mathbf{u}_k^{(k-1)} &= \mathbf{u}_k^{(k-2)} - \mathrm{proj}_{\mathbf{u}_{k-1}} \, (\mathbf{u}_k^{(k-2)}).
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| \end{align} </math>
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| Each step finds a vector <math> \mathbf{u}_k^{(i)} </math> orthogonal to <math> \mathbf{u}_k^{(i-1)} </math>. Thus <math> \mathbf{u}_k^{(i)} </math> is also orthogonalized against any errors introduced in computation of <math> \mathbf{u}_k^{(i-1)} </math>.
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| This method is used in the previous animation, when the intermediate v'<sub>3</sub> vector is used when orthogonalizing the blue vector v<sub>3</sub>.
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| == Algorithm ==
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| The following algorithm implements the stabilized Gram–Schmidt orthonormalization. The vectors '''v'''<sub>1</sub>, …, '''v'''<sub>''k''</sub> are replaced by orthonormal vectors which span the same subspace.
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| : '''for''' ''i'' '''from''' 1 '''to''' ''k'' '''do'''
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| :: <math> \mathbf{v}_i \leftarrow \frac{\mathbf{v}_i}{\|\mathbf{v}_i\|} </math> (''normalize'')
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| :: '''for''' ''j'' '''from''' i+1 '''to''' k '''do'''
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| ::: <math> \mathbf{v}_j \leftarrow \mathbf{v}_j - \mathrm{proj}_{\mathbf{v}_{i}} \, (\mathbf{v}_j) </math> (''remove component in direction'' '''v'''<sub>''i''</sub>)
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| :: '''next j'''
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| : '''next i'''
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| The cost of this algorithm is asymptotically 2''nk''<sup>2</sup> floating point operations, where ''n'' is the dimensionality of the vectors {{harv|Golub|Van Loan|1996|loc=§5.2.8}}.
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| ==Determinant formula==
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| The result of the Gram–Schmidt process may be expressed in a non-recursive formula using [[determinant]]s.
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| :<math> \mathbf{e}_j = \frac{1}{\sqrt{D_{j-1} D_j}} \begin{vmatrix}
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| \langle \mathbf{v}_1, \mathbf{v}_1 \rangle & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
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| \langle \mathbf{v}_1, \mathbf{v}_2 \rangle & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \langle \mathbf{v}_1, \mathbf{v}_{j-1} \rangle & \langle \mathbf{v}_2, \mathbf{v}_{j-1} \rangle & \dots &
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| \langle \mathbf{v}_j, \mathbf{v}_{j-1} \rangle \\
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| \mathbf{v}_1 & \mathbf{v}_2 & \dots & \mathbf{v}_j \end{vmatrix} </math>
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| :<math> \mathbf{u}_j = \frac{1}{D_{j-1} } \begin{vmatrix}
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| \langle \mathbf{v}_1, \mathbf{v}_1 \rangle & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
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| \langle \mathbf{v}_1, \mathbf{v}_2 \rangle & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \langle \mathbf{v}_1, \mathbf{v}_{j-1} \rangle & \langle \mathbf{v}_2, \mathbf{v}_{j-1} \rangle & \dots &
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| \langle \mathbf{v}_j, \mathbf{v}_{j-1} \rangle \\
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| \mathbf{v}_1 & \mathbf{v}_2 & \dots & \mathbf{v}_j \end{vmatrix} </math>
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| where ''D'' <sub>0</sub>=1 and, for ''j'' ≥ 1, ''D <sub>j</sub>'' is the [[Gram determinant]]
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| :<math> D_j = \begin{vmatrix}
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| \langle \mathbf{v}_1, \mathbf{v}_1 \rangle & \langle \mathbf{v}_2, \mathbf{v}_1 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_1 \rangle \\
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| \langle \mathbf{v}_1, \mathbf{v}_2 \rangle & \langle \mathbf{v}_2, \mathbf{v}_2 \rangle & \dots & \langle \mathbf{v}_j, \mathbf{v}_2 \rangle \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \langle \mathbf{v}_1, \mathbf{v}_j \rangle & \langle \mathbf{v}_2, \mathbf{v}_j\rangle & \dots &
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| \langle \mathbf{v}_j, \mathbf{v}_j \rangle \end{vmatrix}. </math>
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| Note that the expression for '''u'''<sub>k</sub> is a "formal" determinant, i.e. the matrix contains both scalars
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| and vectors; the meaning of this expression is defined to be the result of a [[Laplace expansion|cofactor expansion]] along
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| the row of vectors.
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| The determinant formula for the Gram-Schmidt is computationally slower (exponentially slower) than the recursive algorithms described above;
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| it is mainly of theoretical interest.
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| == Alternatives ==
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| Other orthogonalization algorithms use [[Householder transformation]]s or [[Givens rotation]]s. The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the <math>j</math>th orthogonalized vector after the <math>j</math>th iteration, while orthogonalization using [[Householder reflection]]s produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for [[iterative method]]s like the [[Arnoldi iteration]].
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| Yet another alternative is motivated by the use of [[Cholesky decomposition]] for [[Ordinary least squares|inverting the matrix of the normal equations in linear least squares]]. Let <math>\mathbf{V}</math> be a [[full rank|full column rank]] matrix, which columns need to be orthogonalized. The matrix <math>\mathbf{V}^{*} \mathbf{V} </math> is [[Hermitian matrix|Hermitian]] and [[Positive definite matrix|positive definite]], so it can be written as <math> \mathbf{V}^{*} \mathbf{V} = \mathbf{L} \mathbf{L}^{*}, </math> using the [[Cholesky decomposition]]. The lower triangular matrix <math>\mathbf{L} </math> with strictly positive diagonal entries is [[invertible]]. Then columns of the matrix <math>\mathbf{U}= \mathbf{V}(\mathbf{L}^{-1})^{*}</math> are [[orthonormal]] and [[linear span|span]] the same subspace as the columns of the original matrix <math>\mathbf{V}</math>. The explicit use of the product <math>\mathbf{V}^{*} \mathbf{V} </math> makes the algorithm unstable, especially if the product's [[condition number]] is large. Nevertheless, this algorithm is used in practice and implemented in some software packages because of its high efficiency and simplicity.
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| In [[quantum mechanics]] there are several orthogonalization schemes with characteristics better suited for applications than the Gram–Schmidt one. The most important among them are the symmetric and the canonical orthonormalization (see Solivérez & Gagliano).{{Clarify|date=October 2012}}
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| ==References==
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| <references/>
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| * {{Citation | last1=Bau III | first1=David | last2=Trefethen | first2=Lloyd N. | author2-link=Lloyd N. Trefethen | title=Numerical linear algebra | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | isbn=978-0-89871-361-9 | year=1997}}.
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| * {{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations | publisher=Johns Hopkins | edition=3rd | isbn=978-0-8018-5414-9 | year=1996}}.
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| * {{Citation | last1=Greub | first1=Werner | title=Linear Algebra | publisher = Springer | edition=4th |year = 1975}}.
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| * {{Citation | last1=Soliverez | first1=C. E. | last2=Gagliano | first2=E.| title=[http://rmf.smf.mx/pdf/rmf/31/4/31_4_743.pdf Orthonormalization on the plane: a geometric approach] | publisher=Mex. J. Phys. '''31''' (Nº 4), pp. 743‑758 | year=1985}}.
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| ==External links==
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| {{Portal|Mathematics}}
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| * {{springer|title=Orthogonalization|id=p/o070420}}
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| * [http://www.math.hmc.edu/calculus/tutorials/gramschmidt/gramschmidt.pdf Harvey Mudd College Math Tutorial on the Gram-Schmidt algorithm]
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| * [http://jeff560.tripod.com/g.html Earliest known uses of some of the words of mathematics: G] The entry "Gram-Schmidt orthogonalization" has some information and references on the origins of the method.
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| * Demos: [http://www.bigsigma.com/en/demo/gram-schmidt-plane Gram Schmidt process in plane] and [http://www.bigsigma.com/en/demo/gram-schmidt-space Gram Schmidt process in space]
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| * [http://www.math.ucla.edu/~tao/resource/general/115a.3.02f/GramSchmidt.html Gram-Schmidt orthogonalization applet]
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| * [http://www.nag.co.uk/numeric/fl/nagdoc_fl24/html/F05/f05conts.html NAG Gram–Schmidt orthogonalization of n vectors of order m routine]
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| * Proof: [http://planetmath.org/ProofOfGramSchmidtOrthogonalizationProcedure Raymond Puzio, Keenan Kidwell. "proof of Gram-Schmidt orthogonalization algorithm" (version 8). PlanetMath.org.]
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| {{linear algebra}}
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| {{DEFAULTSORT:Gram-Schmidt Process}}
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| [[Category:Linear algebra]]
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| [[Category:Functional analysis]]
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