Gram-Schmidt Calculator

Author: Henrick Yau

- 4 October 2026

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The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space. This calculator converts any set of linearly independent vectors into an orthogonal or orthonormal basis.

Vector Input

Vector Dimension:

Select the dimension of your vectors

Number of Vectors:

Select how many vectors to orthogonalize

Calculation Options

Output Type:

Select whether to normalize the output vectors

Decimal Places:

Round results to this many decimal places

Advanced Settings

Inner Product Type:

Select the type of inner product to use

Show calculation steps

Show matrix representation

About the Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of linearly independent vectors into a set of orthogonal or orthonormal vectors. This orthogonal or orthonormal set spans the same subspace as the original set of vectors.

The Algorithm

For a set of linearly independent vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \), the Gram-Schmidt process constructs an orthogonal set \( \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \) as follows:

\begin{align} \mathbf{u}_1 &= \mathbf{v}_1 \\ \mathbf{u}_2 &= \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) \\ \mathbf{u}_3 &= \mathbf{v}_3 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3) \\ \vdots \\ \mathbf{u}_k &= \mathbf{v}_k - \sum_{j=1}^{k-1} \text{proj}_{\mathbf{u}_j}(\mathbf{v}_k) \end{align}

where \( \text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\langle\mathbf{v}, \mathbf{u}\rangle}{\langle\mathbf{u}, \mathbf{u}\rangle} \mathbf{u} \) is the projection of \( \mathbf{v} \) onto \( \mathbf{u} \).

To obtain an orthonormal basis, each vector \( \mathbf{u}_i \) is normalized by dividing by its length: \( \mathbf{e}_i = \frac{\mathbf{u}_i}{|\mathbf{u}_i|} \).

Applications

Linear Algebra: Creating orthogonal bases for vector spaces
Numerical Analysis: QR factorization of matrices
Statistics: Orthogonalizing predictors in regression analysis
Physics: Finding normal modes of oscillation
Signal Processing: Creating orthogonal basis functions
Quantum Mechanics: Constructing orthonormal basis states

Properties of Orthogonal Vectors

Inner Product: For orthogonal vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( \langle\mathbf{u}, \mathbf{v}\rangle = 0 \)
Linear Independence: Non-zero orthogonal vectors are automatically linearly independent
Pythagoras Theorem: \( |\mathbf{u} + \mathbf{v}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 \) for orthogonal vectors
Representation: Any vector in the span can be easily represented as a linear combination

Disclaimer: This calculator provides an implementation of the Gram-Schmidt process for educational purposes. For very large dimensions or ill-conditioned vectors, numerical stability might be affected. In professional applications, consider using specialized numerical libraries. The Gram-Schmidt process may produce inaccurate results with nearly linearly dependent vectors due to floating-point arithmetic limitations.

Gram-Schmidt Orthogonalisation Formula:

Given a set of linearly independent vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \), the orthogonal set \( \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \) is constructed as:

\[ \begin{align*} \mathbf{u}_1 &= \mathbf{v}_1 \\ \mathbf{u}_2 &= \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) \\ \mathbf{u}_3 &= \mathbf{v}_3 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3) \\ \vdots \\ \mathbf{u}_k &= \mathbf{v}_k - \sum_{j=1}^{k-1} \text{proj}_{\mathbf{u}_j}(\mathbf{v}_k) \end{align*} \]

with the projection defined as: \[ \text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\langle\mathbf{v}, \mathbf{u}\rangle}{\langle\mathbf{u}, \mathbf{u}\rangle} \mathbf{u} \]

What the Gram-Schmidt calculator does

The Gram-Schmidt Calculator is an interactive tool that helps you convert a set of linearly independent vectors into an orthogonal or orthonormal basis. This is useful for simplifying complex vector operations and working efficiently in higher-dimensional spaces.

This tool supports both the standard dot product and weighted inner products, offering flexibility for various mathematical or engineering contexts.

Why orthogonalise vectors with this tool

The calculator is particularly helpful when you want to:

Create orthogonal or orthonormal bases for vector spaces
Understand QR decomposition, a foundational process in linear algebra and numerical analysis
Quickly verify the orthogonality of vectors
Apply vector projection in physics, data analysis, or machine learning

It complements other tools such as the QR Factorisation Calculator, Matrix Inverse Calculator, and Vector Projection Calculator by preparing data in a structured, orthogonal format.

Steps to perform Gram-Schmidt orthogonalisation

Follow these steps to perform a Gram-Schmidt process:

Select the dimension of your vectors (e.g., 2D, 3D, etc.).
Choose how many vectors you want to include (up to 5).
Enter each vector's components. Default values are provided for quick testing.
Pick Orthogonal or Orthonormal as the output type.
Optional: adjust decimal precision or select a weighted dot product if needed.
Click "Calculate Gram-Schmidt" to see the results, including:
- Orthogonalised vectors
- Step-by-step breakdowns
- Matrix representations
- Orthogonality checks
- Application tips

Who benefits from vector orthogonalisation

This tool is ideal for:

Students learning about linear independence, vector spaces, or matrix decomposition
Engineers and scientists working on simulations, signal processing, or structural analysis
Data analysts applying matrix transformations in machine learning workflows
Anyone using tools like the LU Decomposition Calculator or Vector Addition Calculator to handle vectors or matrices

Common Questions (FAQ)

What does "orthogonal" mean?

Orthogonal vectors are at right angles to each other. Their inner product is zero, which simplifies many calculations.

What’s the difference between orthogonal and orthonormal?

Orthonormal vectors are orthogonal and each has a length of 1. They are commonly used to define coordinate systems and simplify projections.

Why does the calculator need linearly independent vectors?

If your vectors are not linearly independent, the Gram-Schmidt process cannot produce a valid basis because some vectors can be written as combinations of others.

What’s the use of the weighted inner product?

Weighted inner products are used when different dimensions have different importance or scaling—common in physics or applied mathematics.

How is this related to QR decomposition?

The output of this calculator forms the "Q" matrix in the QR factorisation process, which is often used to solve systems of linear equations.

Helpful Related Tools

Explore other matrix and vector tools that complement Gram-Schmidt calculations:

QR Factorisation Calculator — Orthogonal-triangular decomposition for solving linear systems
LU Decomposition Calculator — Break down matrices into lower and upper components
Vector Projection Calculator — Find projections along directions
Matrix Inverse Calculator — Compute inverses of square matrices
Vector Addition Calculator — Perform basic vector operations

Key uses of the Gram-Schmidt process

The Gram-Schmidt Calculator offers a clear and practical way to turn linearly independent vectors into orthogonal or orthonormal sets. It helps with learning, teaching, and applying vector space transformations. Whether you're analysing data, solving equations, or preparing matrices for further decomposition, this tool adds precision and clarity to your work.