Gramâ€“Schmidt process

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In mathematics and numerical analysis, the Gram–Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rⁿ. Orthogonalization in this context means the following: we start with vectors v₁,…, v_k which are linearly independent and we want to find mutually orthogonal vectors u₁, …, u_k which generate the same subspace as the vectors v₁, …, v_k.

The method is named for 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.

The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).

1 The Gram–Schmidt process
2 Example
3 Numerical stability
4 Algorithm
5 Alternatives
6 References
7 External links

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The Gram–Schmidt process

We define the projection operator by

<math>\mathrm{proj}_{\mathbf{u}}\,\mathbf{v} = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u}. </math>

It projects the vector v orthogonally onto the vector u.

The Gram–Schmidt process then works as follows: Image:Gram–Schmidt process.svg

<math>\mathbf{u}_1 = \mathbf{v}_1,</math>	<math>\mathbf{e}_1 = {\mathbf{u}_1 \over	\mathbf{u}_1	}</math>
<math>\mathbf{u}_2 = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_2, </math>	<math>\mathbf{e}_2 = {\mathbf{u}_2 \over	\mathbf{u}_2	}</math>
<math>\mathbf{u}_3 = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_2}\,\mathbf{v}_3, </math>	<math>\mathbf{e}_3 = {\mathbf{u}_3 \over	\mathbf{u}_3	}</math>
<math>\vdots</math>	<math>\vdots</math>
<math>\mathbf{u}_k = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,\mathbf{v}_k, </math>	<math>\mathbf{e}_k = {\mathbf{u}_k\over	\mathbf{u}_k	}</math>

The sequence u₁, …, u_k is the required system of orthogonal vectors, and the normalized vectors e₁, …, e_k form an orthonormal system.

To check that these formulas yield an orthogonal sequence, first compute ⟨u₁, u₂⟩ by substituting the above formula for u₂: you will get zero. Then use this to compute ⟨u₁, u₃⟩ again by substituting the formula for u₃: you will get zero. The general proof proceeds by mathematical induction.

Geometrically, this method proceeds as follows: to compute u_i, it projects v_i orthogonally onto the subspace U generated by u₁, …, u_i−1, which is the same as the subspace generated by v₁, …, v_i−1. The vector u_i is then defined to be the difference between v_i and this projection, guaranteed to be orthogonal to all of the vectors in the subspace U.

The Gram–Schmidt process also applies to a linearly independent infinite sequence {v_i}_i. The result is an orthogonal (or orthonormal) sequence {u_i}_i such that for natural number n: the algebraic span of v₁, …, v_n is the same as that of u₁, …, u_n.

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Example

Consider the following set of vectors in R² (with the conventional inner product)

<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>

Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors:

<math>\mathbf{u}_1=\mathbf{v}_1=\begin{pmatrix}3\\1\end{pmatrix}</math>

<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>

We check that the vectors u₁ and u₂ are indeed orthogonal:

<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>

We can then normalize the vectors by dividing out their sizes as shown above:

<math>\mathbf{e}_1 = {1 \over \sqrt {10}}\begin{pmatrix}3\\1\end{pmatrix}</math>

<math>\mathbf{e}_2 = {1 \over \sqrt{40 \over 25}} \begin{pmatrix}-2/5\\6/5\end{pmatrix}

= {1\over\sqrt{10}} \begin{pmatrix}-1\\3\end{pmatrix}. </math>

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Numerical stability

When this process is implemented on a computer, then the vectors u_k are not quite orthogonal because of rounding errors. For the Gram–Schmidt process as described above this loss of orthogonality is particularly bad; therefore, it is said that the (naive) Gram–Schmidt process is numerically unstable.

The Gram–Schmidt process can be stabilized by a small modification. Instead of computing the vector u_k as

<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

<math> \mathbf{u}_k^{(1)} = \mathbf{v}_k - \mathrm{proj}_{\mathbf{u}_1}\,\mathbf{v}_k, </math>

<math> \mathbf{u}_k^{(2)} = \mathbf{u}_k^{(1)} - \mathrm{proj}_{\mathbf{u}_2} \, \mathbf{u}_k^{(1)}, </math>

<math> \vdots </math>

<math> \mathbf{u}_k^{(k-2)} = \mathbf{u}_k^{(k-3)} - \mathrm{proj}_{\mathbf{u}_{k-2}} \, \mathbf{u}_k^{(k-3)}, </math>

<math> \mathbf{u}_k = \mathbf{u}_k^{(k-2)} - \mathrm{proj}_{\mathbf{u}_{k-1}} \, \mathbf{u}_k^{(k-2)}. </math>

This series of computations gives the same result as the original formula in exact arithmetic, but it introduces smaller errors in finite-precision arithmetic.

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Algorithm

The following algorithm implements the stabilized Gram–Schmidt process. The vectors v₁, …, v_k are replaced by orthonormal vectors which span the same subspace.

for j from 1 to k do

for i from 1 to j − 1 do

<math> \mathbf{v}_j \leftarrow \mathbf{v}_j - \langle \mathbf{v}_j, \mathbf{v}_i \rangle \mathbf{v}_i </math> (remove component in direction v_i)

end for

<math> \mathbf{v}_j \leftarrow \frac{\mathbf{v}_j}{\|\mathbf{v}_j\|} </math> (normalize)

end for

The cost of this algorithm is asymptotically 2kn² floating point operations, where n is the size of the vectors.

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Alternatives

Other orthogonalization algorithms use Householder transformations or Givens rotations. The algorithms using Householder transformations is more stable than the (stabilized) Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration

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