Gram-Schmidt Orthogonalization

Compute orthogonal¹ and orthonormal² bases using the Gram-Schmidt process³.
Enter matrix dimensions \(m \times n\), generate input matrix \(A\), fill values, and calculate. Columns of \(A\) are the vectors. If linearly dependent, a subset will be used.

The Gram-Schmidt process converts a set of linearly independent vectors into an orthogonal (or orthonormal) set that spans the same subspace.

Algorithm:

Starting with vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\):

\(\mathbf{u}_1 = \mathbf{v}_1\)
\(\mathbf{u}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} \frac{\mathbf{v}_k \cdot \mathbf{u}_j}{\mathbf{u}_j \cdot \mathbf{u}_j} \mathbf{u}_j\) (orthogonal vectors)
\(\mathbf{e}_k = \frac{\mathbf{u}_k}{\|\mathbf{u}_k\|}\) (orthonormal vectors)

Applications:

QR decomposition: \(A = QR\) where \(Q\) has orthonormal columns
Solving least squares problems
Finding orthonormal bases for subspaces

This calculator uses exact arithmetic via a custom Fraction class and displays results using surd notation⁴ when possible.

Rows \(m\): Columns \(n\):

Enter dimensions and click Generate Matrix to create the input grid.

1 — Set Up

Loads a pre-filled 3×3 matrix — ready to orthogonalize.

2 — Compute

Applies the Gram–Schmidt process step-by-step to produce orthogonal and orthonormal bases.

Resets all dimensions, matrix entries, and results.

Footnotes

An orthogonal basis is a set of mutually perpendicular vectors that span a subspace. Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal if their inner product \(\langle \mathbf{u}, \mathbf{v} \rangle = 0\). Orthogonal bases simplify projections and decompositions. ↑
An orthonormal basis is an orthogonal basis where every vector has unit length: \(\|\mathbf{e}_k\| = 1\). Orthonormal bases are especially convenient because the projection of a vector \(\mathbf{v}\) onto a basis vector \(\mathbf{e}_k\) is simply \(\langle \mathbf{v}, \mathbf{e}_k \rangle \mathbf{e}_k\). ↑
The Gram-Schmidt process is an algorithm that takes a finite set of linearly independent vectors and generates an orthogonal (or orthonormal) set spanning the same subspace. It proceeds iteratively, subtracting the projections of each new vector onto all previously computed orthogonal vectors. ↑
The inner product (or dot product) of two vectors \(\mathbf{u} = (u_1, \ldots, u_m)\) and \(\mathbf{v} = (v_1, \ldots, v_m)\) is \(\langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^{m} u_i v_i\). It measures the degree to which two vectors point in the same direction and is fundamental to the Gram-Schmidt process. ↑