Gram-Schmidt Calculator

Welcome to the Gram-Schmidt Calculator, a comprehensive linear algebra tool that orthonormalizes a set of linearly independent vectors using the classical Gram-Schmidt process. Get detailed step-by-step projections, both orthogonal and orthonormal bases, interactive vector visualization, and orthogonality verification. Ideal for students, educators, engineers, and anyone working with vector spaces.

What is the Gram-Schmidt Process?

The Gram-Schmidt process (named after Jørgen Pedersen Gram and Erhard Schmidt) is a method for orthonormalizing a set of vectors in an inner product space. Given a set of linearly independent vectors \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\), the process produces an orthonormal set \(\{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\}\) that spans the same subspace.

The Algorithm

The Gram-Schmidt process works in two phases for each vector:

1. Orthogonalization: Subtract the projections onto all previously computed orthogonal vectors
2. Normalization: Divide by the norm to get a unit vector

Where \(\langle \mathbf{a}, \mathbf{b} \rangle\) denotes the inner (dot) product and \(\|\mathbf{u}\| = \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle}\) is the Euclidean norm.

How to Use This Calculator

1. Enter your vectors: Input linearly independent vectors, one per line. Use parentheses, brackets, or just comma-separated values. All vectors must have the same dimension (2 to 10).
2. Set decimal precision: Choose how many decimal places (2-10) to display in results.
3. Click Orthonormalize: The calculator performs the full Gram-Schmidt process and shows complete results.
4. Review results: Examine the orthonormal basis, interactive visualization, step-by-step projections, and orthogonality verification.

Understanding the Results

Orthogonal Basis (\(\mathbf{u}_k\))

The intermediate orthogonal vectors before normalization. These vectors are mutually perpendicular but may have different magnitudes. The orthogonal basis preserves the integer/rational structure of the original vectors, which is sometimes preferred in theoretical work.

Orthonormal Basis (\(\mathbf{e}_k\))

The final output — vectors that are both mutually perpendicular (orthogonal) and have unit length (normal). This is the standard output of the Gram-Schmidt process and the most commonly used form.

Verification Table

The calculator verifies orthonormality by computing all pairwise dot products (which should be 0 for distinct pairs) and all norms (which should be 1). This serves as a mathematical proof that the process succeeded.

Connection to QR Decomposition

The Gram-Schmidt process is the classical method for computing the QR decomposition of a matrix. If you arrange the input vectors as columns of matrix \(A\) and the orthonormal vectors as columns of matrix \(Q\), then:

Where \(Q\) is an orthogonal matrix (its columns are the orthonormal vectors) and \(R\) is upper triangular (its entries are the projection coefficients). QR decomposition is fundamental in numerical linear algebra for solving least-squares problems, computing eigenvalues, and matrix factorization.

Applications

Classical vs Modified Gram-Schmidt

This calculator implements the classical Gram-Schmidt (CGS) algorithm. For numerical computations with floating-point arithmetic, the Modified Gram-Schmidt (MGS) algorithm offers better numerical stability by recomputing projections against the partially orthogonalized set rather than the original vectors. However, in exact arithmetic (or high-precision computation), both algorithms produce identical results.

Frequently Asked Questions

What is the Gram-Schmidt process?

The Gram-Schmidt process is an algorithm for orthonormalizing a set of vectors in an inner product space. It takes a set of linearly independent vectors and produces an orthonormal set that spans the same subspace. Each vector is made orthogonal to all previous vectors by subtracting its projections, then normalized to unit length.

Why is the Gram-Schmidt process important?

The Gram-Schmidt process is fundamental in linear algebra and has many applications: QR decomposition of matrices, solving least-squares problems, constructing orthonormal bases for function spaces (e.g., Legendre polynomials), signal processing, computer graphics, and numerical methods. Orthonormal bases simplify many calculations because the basis vectors are perpendicular and have unit length.

What is the difference between orthogonal and orthonormal vectors?

Orthogonal vectors are perpendicular to each other (their dot product is zero), but they can have any magnitude. Orthonormal vectors are both orthogonal AND have unit length (magnitude = 1). The Gram-Schmidt process first makes vectors orthogonal, then normalizes them to produce an orthonormal set.

What happens if the input vectors are linearly dependent?

If the input vectors are linearly dependent, the Gram-Schmidt process will produce a zero vector at some step (when a vector lies in the span of previous vectors). This calculator detects linear dependence and reports an error. To use this calculator, all input vectors must be linearly independent.

How does Gram-Schmidt relate to QR decomposition?

QR decomposition factors a matrix A into Q (orthogonal matrix) and R (upper triangular matrix). The Gram-Schmidt process applied to the columns of A produces the columns of Q, while the projection coefficients form the entries of R. This connection makes Gram-Schmidt the classical method for computing QR factorization.

Additional Resources

• Gram-Schmidt Process - Wikipedia
• QR Decomposition - Wikipedia
• Orthonormality - Wikipedia