Gram-Schmidt Calculator

Category: Linear Algebra
- August 14, 2025
| |

The Gram-Schmidt process is a method for orthogonalising 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

Select the dimension of your vectors
Select how many vectors to orthogonalise

Calculation Options

Select whether to normalise the output vectors
Round results to this many decimal places

Advanced Settings

Select the type of inner product to use

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 normalised 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 factorisation of matrices
  • Statistics: Orthogonalising 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 specialised numerical libraries. The Gram-Schmidt process may produce inaccurate results with nearly linearly dependent vectors due to floating-point arithmetic limitations.

Gram-Schmidt Orthogonalization 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 Is the Gram-Schmidt Calculator?

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 standard dot product and weighted inner products, giving flexibility for different mathematical or engineering contexts.

Why Use This Tool?

The calculator is especially 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
  • Verify orthogonality of vectors quickly
  • Apply vector projection in Physics, data analysis, or machine learning

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

How to Use the Calculator

Follow these steps to perform a Gram-Schmidt process:

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

Who Can Benefit?

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 factorization 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:

Summary

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.