Gram–Schmidt process

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space R^"n". The Gram–Schmidt process takes a finite, linearly independent set "S" = {"v"₁, …, "v"_"n"} and generates an orthogonal set "S"' = {"u"₁, …, "u"_"n"} that spans the same subspace as "S".

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).

The Gram–Schmidt process

We define the projection operator by:$$mathrm{proj}_{mathbf{u,mathbf{v} = {langle mathbf{u}, mathbf{v} angleoverlangle mathbf{u}, mathbf{u} angle}mathbf{u} = {langle mathbf{u}, mathbf{v} angle} {mathbf{u}overlangle mathbf{u}, mathbf{u} angle}, where <u, v> denotes the inner product of the vectors u and v. This operator projects the vector v orthogonally onto the vector u.

The Gram–Schmidt process then works as follows:

-
| ||$$mathbf{u}_2 = mathbf{v}_2-mathrm{proj}_{mathbf{u}_1},mathbf{v}_2,
| ||$$mathbf{e}_2 = {mathbf{u}_2 over |mathbf{u}_2
-
| ||$$mathbf{u}_3 = mathbf{v}_3-mathrm{proj}_{mathbf{u}_1},mathbf{v}_3-mathrm{proj}_{mathbf{u}_2},mathbf{v}_3,
| ||$$mathbf{e}_3 = {mathbf{u}_3 over |mathbf{u}_3
-
| ||$$mathbf{u}_4 = mathbf{v}_4-mathrm{proj}_{mathbf{u}_1},mathbf{v}_4-mathrm{proj}_{mathbf{u}_2},mathbf{v}_4-mathrm{proj}_{mathbf{u}_3},mathbf{v}_4,
| ||$$mathbf{e}_4 = {mathbf{u}_4 over |mathbf{u}_4
-
| ||align="center"|$$vdots
| ||align="center"|$$vdots
-
| ||$$mathbf{u}_k = mathbf{v}_k-sum_{j=1}^{k-1}mathrm{proj}_{mathbf{u}_j},mathbf{v}_k,
| ||$$mathbf{e}_k = {mathbf{u}_kover |mathbf{u}_k The sequence u₁, &hellip;, u_"k" is the required system of orthogonal vectors, and the normalized vectors e₁, &hellip;, e_"k" form an ortho"normal" set.

To check that these formulas yield an orthogonal sequence, first compute &lang;u₁, u₂&rang; by substituting the above formula for u₂: you will get zero. Then use this to compute &lang;u₁, u₃&rang; 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, &hellip;, u_"i"−1, which is the same as the subspace generated by v₁, &hellip;, 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, &hellip;, v_"n" is the same as that of u₁, &hellip;, u_"n".

If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the $$ith step, assuming that $$mathbf{v_i} is a linear combination of $$mathbf{v_1}, mathbf{v_2}, ldots, mathbf{v_{i-1.

Example

Consider the following set of vectors in R² (with the conventional inner product):$$S = leftlbracemathbf{v}_1=egin{pmatrix} 3 \ 1end{pmatrix}, mathbf{v}_2=egin{pmatrix}2 \2end{pmatrix} ight brace.

Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors::$$mathbf{u}_1=mathbf{v}_1=egin{pmatrix}3\1end{pmatrix}:$$mathbf{u}_2 = mathbf{v}_2 - mathrm{proj}_{mathbf{u}_1} , mathbf{v}_2 = egin{pmatrix}2\2end{pmatrix} - mathrm{proj}_{({3 atop 1})} , {egin{pmatrix}2\2end{pmatrix = egin{pmatrix} -2/5 \6/5 end{pmatrix}.

We check that the vectors u₁ and u₂ are indeed orthogonal::$$langlemathbf{u}_1,mathbf{u}_2 angle = leftlangle egin{pmatrix}3\1end{pmatrix}, egin{pmatrix}-2/5\6/5end{pmatrix} ight angle = -frac65 + frac65 = 0.

We can then normalize the vectors by dividing out their sizes as shown above::$$mathbf{e}_1 = {1 over sqrt {10egin{pmatrix}3\1end{pmatrix}:$$mathbf{e}_2 = {1 over sqrt{40 over 25 egin{pmatrix}-2/5\6/5end{pmatrix} = {1oversqrt{10 egin{pmatrix}-1\3end{pmatrix}.

Numerical stability

When this process is implemented on a computer, the vectors $$u_k are often not quite orthogonal, due to rounding errors. For the Gram–Schmidt process as described above this loss of orthogonality is particularly bad; therefore, it is said that the (na&iuml;ve) 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:$$mathbf{u}_k = mathbf{v}_k - mathrm{proj}_{mathbf{u}_1},mathbf{v}_k - mathrm{proj}_{mathbf{u}_2},mathbf{v}_k - cdots - mathrm{proj}_{mathbf{u}_{k-1,mathbf{v}_k, it is computed as:$$mathbf{u}_k^{(1)} = mathbf{v}_k - mathrm{proj}_{mathbf{u}_1},mathbf{v}_k, :$$mathbf{u}_k^{(2)} = mathbf{u}_k^{(1)} - mathrm{proj}_{mathbf{u}_2} , mathbf{u}_k^{(1)}, :::$$vdots :$$mathbf{u}_k^{(k-2)} = mathbf{u}_k^{(k-3)} - mathrm{proj}_{mathbf{u}_{k-2 , mathbf{u}_k^{(k-3)}, :$$mathbf{u}_k = mathbf{u}_k^{(k-2)} - mathrm{proj}_{mathbf{u}_{k-1 , mathbf{u}_k^{(k-2)}. This series of computations gives the same result as the original formula in exact arithmetic, but it introduces smaller errors in finite-precision arithmetic.

Algorithm

The following algorithm implements the stabilized Gram–Schmidt process. The vectors v₁, &hellip;, 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::: $$mathbf{v}_j leftarrow mathbf{v}_j - mathrm{proj}_{mathbf{v}_{i , mathbf{v}_j ("remove component in direction" v_"i"):: end for:: $$mathbf{v}_j leftarrow frac{mathbf{v}_j}{|mathbf{v}_j ("normalize"): end forThe cost of this algorithm is asymptotically 2"nk"² floating point operations, where "n" is the dimensionality of the vectors harv|Golub|Van Loan|1996|loc=§5.2.8.

Alternatives

Other orthogonalization algorithms use Householder transformations or Givens rotations. The algorithms using Householder transformations are 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.

References

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External links

* [http://video.google.com/videoplay?docid=-2274442444593051761 MIT Linear Algebra Lecture on Orthogonal Matrices] at Google Video, from MIT OpenCourseWare
* [http://members.aol.com/jeff570/g.html Gram–Schmidt orthogonalization] on the [http://members.aol.com/jeff570/mathword.html Earliest known uses of some of the words of mathematics]