In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. 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 after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt.^[1] 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).

We define the projection operator by

The Gram–Schmidt process then works as follows:

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

Geometrically, this method proceeds as follows: to compute ui, it projects vi orthogonally onto the subspace U generated by u1, ..., ui−1, which is the same as the subspace generated by v1, ..., vi−1. The vector ui is then defined to be the difference between vi 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 countably infinite sequence {vi}i. The result is an orthogonal (or orthonormal) sequence {ui}i such that for natural number n: the algebraic span of v1, ..., vn is the same as that of u1, ..., un.

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

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

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

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

noting that if the dot product of two vectors is 0 then they are orthogonal.

For non-zero vectors, we can then normalize the vectors by dividing out their sizes as shown above:

It has the following properties:

The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS. This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic. Instead of computing the vector uk as

it is computed as

This method is used in the previous animation, when the intermediate v'3 vector is used when orthogonalizing the blue vector v3.

The following MATLAB algorithm implements the stabilized Gram–Schmidt orthonormalization for Euclidean Vectors. The vectors v1, ..., vk (columns of matrix V, so that V(:,j) is the jth vector) are replaced by orthonormal vectors (columns of U) which span the same subspace.

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

And reducing this to row echelon form produces

The normalized vectors are then

as in the example above.

The result of the Gram–Schmidt process may be expressed in a non-recursive formula using determinants.

where D 0=1 and, for j ≥ 1, D j is the Gram determinant

Note that the expression for uk is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a cofactor expansion along the row of vectors.

The determinant formula for the Gram-Schmidt is computationally slower (exponentially slower) than the recursive algorithms described above; it is mainly of theoretical interest.

In quantum mechanics there are several orthogonalization schemes with characteristics better suited for certain applications than original Gram–Schmidt. Nevertheless, it remains a popular and effective algorithm for even the largest electronic structure calculations.^[3]