Gram-Schmidt calculator



Warning: Undefined variable $table in /home/mxncalc/public_html/matrix-calculator/modelv.php on line 13
Matrices

📥 Export options

Exporting...

⭐ Rate this tool:

★ ★ ★ ★ ★

⭐ 3.0/5 (2 votes)



Transform any set of linearly independent vectors into an orthogonal or orthonormal basis with detailed step-by-step solutions

The Gram-Schmidt process is a fundamental algorithm in linear algebra that converts a set of linearly independent vectors into an orthogonal (perpendicular) or orthonormal (perpendicular with unit length) set. Our calculator performs both orthogonalization and orthonormalization, showing you every step of the transformation.

Quick Navigation

What is Gram-Schmidt? Orthogonal vs Orthonormal The Algorithm Detailed Example Geometric View QR Decomposition Applications FAQ

What is the Gram-Schmidt Process?

Definition

The Gram-Schmidt process is an algorithm that takes a set of linearly independent vectors {v₁, v₂, ..., vₙ} and produces an orthogonal set {u₁, u₂, ..., uₙ} that spans the same subspace. The process works by:

  • Taking the first vector as is
  • Making each subsequent vector orthogonal to all previous ones
  • Optionally normalizing to create unit vectors (orthonormalization)
u₁ = v₁
u₂ = v₂ - proj_{u₁}(v₂)
u₃ = v₃ - proj_{u₁}(v₃) - proj_{u₂}(v₃)
...

Key Concepts

📐 Orthogonality

Two vectors are orthogonal if their dot product is zero: u·v = 0. This means they're perpendicular in space.

🎯 Projection

The projection of vector v onto u is: proj_u(v) = ((v·u)/(u·u)) × u

📏 Normalization

A normalized vector has length 1. We normalize by dividing by the vector's magnitude: û = u/||u||

🔄 Basis Preservation

The resulting vectors span exactly the same space as the original vectors.

Orthogonal vs Orthonormal Bases

Property Orthogonal Basis Orthonormal Basis
Perpendicularity All vectors perpendicular (u_i · u_j = 0 for i≠j) All vectors perpendicular (same condition)
Vector Length Can be any non-zero length All vectors have unit length (||u_i|| = 1)
Dot Product Matrix Diagonal matrix Identity matrix
Computation Gram-Schmidt without normalization Gram-Schmidt with normalization
Use Cases General orthogonal transformations QR decomposition, rotations, reflections
Pro Tip: Orthonormal bases are especially useful because they preserve lengths and angles under transformation, making calculations much simpler!

The Gram-Schmidt Algorithm

Classical Gram-Schmidt Process

Algorithm Steps
  1. Initialize: Set u₁ = v₁
  2. For each vector vₖ (k = 2 to n):
    • Calculate projections onto all previous orthogonal vectors
    • Subtract all projections from vₖ
    • Result is uₖ, orthogonal to all previous vectors
  3. Optional - Normalize: For orthonormal basis, divide each uₖ by its magnitude

Mathematical Formulation

For k = 1 to n:
    u_k = v_k
    For j = 1 to k-1:
        u_k = u_k - proj_{u_j}(v_k)
        where proj_{u_j}(v_k) = (v_k · u_j)/(u_j · u_j) × u_j
    
    If orthonormalization required:
        e_k = u_k / ||u_k||

Modified Gram-Schmidt (More Stable)

Numerical Stability

The modified Gram-Schmidt process reorthogonalizes at each step, providing better numerical stability for computer calculations:

  • Instead of subtracting all projections at once, subtract them one by one
  • Update the working vector after each projection subtraction
  • Reduces accumulation of rounding errors

Step-by-Step Example: 3D Vectors

Complete Gram-Schmidt Orthonormalization

Given Vectors:

v₁ = [3, 1, 0]
v₂ = [2, 2, 0]
v₃ = [1, 1, 1]

Step 1 First Orthogonal Vector

u₁ = v₁ = [3, 1, 0]

Step 2 Second Orthogonal Vector

Calculate projection of v₂ onto u₁:

proj_{u₁}(v₂) = ((v₂·u₁)/(u₁·u₁)) × u₁
                = ((2×3 + 2×1 + 0×0)/(3² + 1² + 0²)) × [3, 1, 0]
                = (8/10) × [3, 1, 0] = [2.4, 0.8, 0]
u₂ = v₂ - proj_{u₁}(v₂) = [2, 2, 0] - [2.4, 0.8, 0] = [-0.4, 1.2, 0]

Step 3 Third Orthogonal Vector

Calculate projections onto u₁ and u₂:

proj_{u₁}(v₃) = ((4)/(10)) × [3, 1, 0] = [1.2, 0.4, 0]
proj_{u₂}(v₃) = ((0.8)/(1.6)) × [-0.4, 1.2, 0] = [-0.2, 0.6, 0]
u₃ = v₃ - proj_{u₁}(v₃) - proj_{u₂}(v₃) = [0, 0, 1]

Step 4 Normalize (for Orthonormal Basis)

e₁ = u₁/||u₁|| = [3, 1, 0]/√10 ≈ [0.949, 0.316, 0]
e₂ = u₂/||u₂|| = [-0.4, 1.2, 0]/√1.6 ≈ [-0.316, 0.949, 0]
e₃ = u₃/||u₃|| = [0, 0, 1]/1 = [0, 0, 1]
Verification: Check that e₁·e₂ = 0, e₁·e₃ = 0, e₂·e₃ = 0, and ||e₁|| = ||e₂|| = ||e₃|| = 1

Geometric Interpretation

Visual Understanding

The Gram-Schmidt process can be visualized as:

  • Step 1: Take the first vector as your first axis direction
  • Step 2: Find the component of the second vector perpendicular to the first
  • Step 3: Find the component of the third vector perpendicular to the plane of the first two
  • Continue: Each new vector is perpendicular to the subspace of all previous ones

🌐 Subspace Preservation

At each step k, the vectors {u₁, ..., uₖ} span the same subspace as {v₁, ..., vₖ}

📊 Dimension Reduction

Each new orthogonal vector explores a new dimension perpendicular to all previous ones

🎨 Coordinate System

The result forms a new coordinate system where all axes are perpendicular

🔄 Rotation Interpretation

Orthonormal bases represent pure rotations (and possibly reflections) of the standard basis

Connection to QR Decomposition

QR Factorization

The Gram-Schmidt process directly produces the QR decomposition of a matrix:

A = QR

where Q is orthogonal (orthonormal columns) and R is upper triangular.

How It Works

From Gram-Schmidt to QR
  • Matrix A: Original vectors as columns [v₁ | v₂ | ... | vₙ]
  • Matrix Q: Orthonormalized vectors as columns [e₁ | e₂ | ... | eₙ]
  • Matrix R: Upper triangular matrix with:
    • Diagonal entries: R_ii = ||u_i|| (lengths before normalization)
    • Upper entries: R_ij = e_i · v_j (projections)
    • Lower entries: All zeros
Application: QR decomposition is fundamental in solving least squares problems, eigenvalue algorithms, and numerical linear algebra.

Applications of Gram-Schmidt Process

Mathematical Applications

📐 QR Decomposition

Direct method for QR factorization, essential for solving linear systems and eigenvalue problems

📊 Least Squares

Solving overdetermined systems and finding best-fit solutions through orthogonal projections

🔢 Numerical Stability

Creating stable bases for numerical computations in finite-precision arithmetic

Real-World Applications

🎯 Signal Processing

Creating orthogonal signal bases, wavelet transforms, and Fourier analysis foundations

🖼️ Computer Graphics

Constructing orthonormal coordinate frames, camera matrices, and view transformations

🤖 Machine Learning

Feature decorrelation, whitening transformations, and independent component analysis

🛰️ Communications

CDMA systems, orthogonal frequency-division multiplexing (OFDM), and channel coding

⚛️ Quantum Computing

Creating orthonormal quantum states and implementing quantum gates

📈 Data Science

Principal Component Analysis (PCA) preparation and multicollinearity handling

Numerical Considerations

Loss of Orthogonality: In finite-precision arithmetic, the classical Gram-Schmidt can suffer from loss of orthogonality due to rounding errors. Use modified Gram-Schmidt or reorthogonalization for better results.
Method Stability Operations When to Use
Classical GS Can be unstable 2mn² flops Small matrices, educational purposes
Modified GS More stable 2mn² flops General purpose, moderate sizes
GS with Reorthogonalization Very stable 4mn² flops High precision requirements
Householder QR Most stable 2mn² - 2n³/3 flops Professional applications

Frequently Asked Questions

What is the Gram-Schmidt process used for?
The Gram-Schmidt process converts any set of linearly independent vectors into an orthogonal or orthonormal set that spans the same space. It's used for QR decomposition, solving least squares problems, creating orthogonal bases for function spaces, signal processing, and many numerical algorithms in linear algebra.
What's the difference between orthogonal and orthonormal?
Orthogonal vectors are perpendicular to each other (their dot product is zero), but can have any non-zero length. Orthonormal vectors are both orthogonal AND have unit length (magnitude = 1). The Gram-Schmidt process produces orthogonal vectors, which can then be normalized to create orthonormal vectors.
Can Gram-Schmidt fail?
The Gram-Schmidt process fails if the input vectors are linearly dependent. When a vector is a linear combination of previous vectors, the subtraction of projections results in the zero vector, which cannot be normalized. Always ensure your input vectors are linearly independent.
How do I verify the result is correct?
Check that: 1) All pairs of resulting vectors have dot product zero (orthogonality), 2) Each vector has length 1 if orthonormalized, 3) The resulting vectors span the same space as the original vectors (they should have the same rank when formed into matrices).
What is modified Gram-Schmidt?
Modified Gram-Schmidt is a numerically more stable variant where projections are computed and subtracted one at a time, updating the working vector after each subtraction. This reduces the accumulation of rounding errors compared to classical Gram-Schmidt, which computes all projections before subtracting.
How is Gram-Schmidt related to QR decomposition?
The Gram-Schmidt process directly produces QR decomposition: the orthonormalized vectors form the columns of Q (orthogonal matrix), and the projection coefficients and norms form R (upper triangular matrix). This gives A = QR where A is the original matrix of vectors.
Can I use Gram-Schmidt for complex vectors?
Yes! For complex vectors, replace the dot product with the inner product (conjugate of first vector dotted with second), and use the complex norm. The process produces a unitary basis instead of orthonormal. This is essential in quantum mechanics and signal processing.
What happens with nearly dependent vectors?
Nearly linearly dependent vectors (small angle between them) can cause numerical instability. The resulting orthogonal vectors may have very small magnitudes, leading to large errors when normalized. Use modified Gram-Schmidt or consider using SVD for better numerical stability in these cases.

Ready to Orthogonalize Your Vectors?

Use our Gram-Schmidt calculator to transform any set of vectors into an orthogonal or orthonormal basis instantly!

Tips for Using Gram-Schmidt Process

Quick Check: Before starting, verify your vectors are linearly independent by checking if their determinant (when arranged as matrix columns) is non-zero.
  • Start with vectors that are already somewhat orthogonal for better numerical stability
  • Normalize at the end rather than at each step to avoid accumulating division errors
  • For hand calculations, work with fractions to maintain exact values
  • Check orthogonality frequently: u_i · u_j should equal zero for i ≠ j
  • Remember that the order of input vectors matters - different orders give different results
  • Use our Gram-Schmidt calculator to verify your manual calculations
  • For large vectors or many dimensions, consider using numerical software
Common Application: Creating an orthonormal basis from eigenvectors is a frequent use case, especially when eigenvectors from numerical algorithms aren't perfectly orthogonal.