orthogonal – Ware

the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly theEuclidean space Rⁿ. The Gram–Schmidt process takes a finite, linearly independent set S = {v₁, …, v_k} for k ≤ n and generates an orthogonal set S′ = {u₁, …, u_k} that spans the same k-dimensional subspace of Rⁿ 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 theIwasawa 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

$mathrm{proj}_{mathbf{u}},(mathbf{v}) = {langle mathbf{v}, mathbf{u}rangleoverlangle mathbf{u}, mathbf{u}rangle}mathbf{u} ,$

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:

$begin{align} mathbf{u}_1 & = mathbf{v}_1, & mathbf{e}_1 & = {mathbf{u}_1 over |mathbf{u}_1|} \ mathbf{u}_2 & = mathbf{v}_2-mathrm{proj}_{mathbf{u}_1},(mathbf{v}_2), & mathbf{e}_2 & = {mathbf{u}_2 over |mathbf{u}_2|} \ mathbf{u}_3 & = mathbf{v}_3-mathrm{proj}_{mathbf{u}_1},(mathbf{v}_3)-mathrm{proj}_{mathbf{u}_2},(mathbf{v}_3), & mathbf{e}_3 & = {mathbf{u}_3 over |mathbf{u}_3|} \ mathbf{u}_4 & = mathbf{v}_4-mathrm{proj}_{mathbf{u}_1},(mathbf{v}_4)-mathrm{proj}_{mathbf{u}_2},(mathbf{v}_4)-mathrm{proj}_{mathbf{u}_3},(mathbf{v}_4), & mathbf{e}_4 & = {mathbf{u}_4 over |mathbf{u}_4|} \ & {} vdots & & {} vdots \ mathbf{u}_k & = mathbf{v}_k-sum_{j=1}^{k-1}mathrm{proj}_{mathbf{u}_j},(mathbf{v}_k), & mathbf{e}_k & = {mathbf{u}_kover |mathbf{u}_k |}. end{align}$

The first two steps of the Gram–Schmidt process

The sequence u₁, …, u_k is the required system of orthogonal vectors, and the normalized vectors e₁, …, e_k form an orthonormal set. The calculation of the sequence u₁, …, u_k is known as Gram–Schmidt orthogonalization, while the calculation of the sequence e₁, …,e_k is known as Gram–Schmidt orthonormalization as the vectors are normalized.

To check that these formulas yield an orthogonal sequence, first compute 〈u₁, u₂〉 by substituting the above formula for u₂: we get zero. Then use this to compute 〈u₁, u₃〉 again by substituting the formula for u₃: we get zero. The general proof proceeds bymathematical 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.

If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the ith step, assuming that v_i is a linear combination of v₁, …, v_i−1. 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.

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 (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 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

$begin{align} mathbf{u}_k^{(1)} &= mathbf{v}_k - mathrm{proj}_{mathbf{u}_1},(mathbf{v}_k), \ mathbf{u}_k^{(2)} &= mathbf{u}_k^{(1)} - mathrm{proj}_{mathbf{u}_2} , (mathbf{u}_k^{(1)}), \ & ,,, vdots \ mathbf{u}_k^{(k-2)} &= mathbf{u}_k^{(k-3)} - mathrm{proj}_{mathbf{u}_{k-2}} , (mathbf{u}_k^{(k-3)}), \ mathbf{u}_k^{(k-1)} &= mathbf{u}_k^{(k-2)} - mathrm{proj}_{mathbf{u}_{k-1}} , (mathbf{u}_k^{(k-2)}). end{align}$

Each step finds a vector $mathbf{u}_k^{(i)}$ orthogonal to $mathbf{u}_k^{(i-1)}$ . Thus $mathbf{u}_k^{(i)}$ is also orthogonalized against any errors introduced in computation of $mathbf{u}_k^{(i-1)}$ . This approach (sometimes referred to as “modified Gram–Schmidt”) gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic.

Algorithm

The following algorithm implements the stabilized Gram–Schmidt orthonormalization. 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

$mathbf{v}_j leftarrow mathbf{v}_j - mathrm{proj}_{mathbf{v}_{i}} , (mathbf{v}_j)$ (remove component in direction v_i)

next i

$mathbf{v}_j leftarrow frac{mathbf{v}_j}{|mathbf{v}_j|}$ (normalize)

next j

The cost of this algorithm is asymptotically 2nk² floating point operations, where n is the dimensionality of the vectors (Golub & Van Loan 1996, §5.2.8)