1. What is Gram-Schmidt Orthogonalization?

Gram-Schmidt orthogonalization is a process that takes a set of vectors and produces an orthogonal set that spans the same subspace. It's fundamental in linear algebra for creating orthogonal bases.

2. How Does the Calculator Work?

The calculator implements the Gram-Schmidt process:

Where:

• \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k \) — Original vectors (matrix columns)
• \( \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_k \) — Orthogonal vectors
• \( \text{proj}_{\mathbf{u}}(\mathbf{v}) \) — Projection of v onto u

Explanation: Each vector is made orthogonal to all previous vectors by subtracting its projection onto them.

3. Importance of Matrix Orthogonalization

Details: Orthogonal matrices are crucial in QR decomposition, least squares problems, eigenvalue computations, and many numerical algorithms.

4. Using the Calculator

Tips: Enter matrix with comma-separated values within rows and semicolon-separated rows. Example: "1,0,1; 1,1,0; 0,1,1" for a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between orthogonal and orthonormal?
A: Orthogonal vectors are perpendicular; orthonormal vectors are orthogonal AND have unit length. Gram-Schmidt produces orthogonal vectors that can be normalized.

Q2: Does Gram-Schmidt work for complex matrices?
A: Yes, but the inner product becomes complex conjugate. This calculator handles real matrices only.

Q3: What if my matrix has linearly dependent columns?
A: Gram-Schmidt will produce zero vectors for dependent columns. The rank equals the number of non-zero output vectors.

Q4: How is this related to QR factorization?
A: Gram-Schmidt computes the Q matrix in QR decomposition, where Q is orthogonal and R is upper triangular.

Q5: What are the numerical stability issues?
A: Classical Gram-Schmidt can lose orthogonality due to rounding errors. Modified Gram-Schmidt is more stable but not implemented here.