Orthonormal Basis

Formula

\[ q_i^T q_j = \delta_{ij},\quad i,j=1,\ldots,n \]

Parameters

• \(\{q_i\}\): basis vectors
• \(\delta_{ij}\): Kronecker delta

What it means

A basis with unit-length, mutually orthogonal vectors.

What it's used for

• Representing vectors without redundancy.
• Simplifying projections and coordinate changes.

Key properties

• Matrix \(Q=[q_1\;\cdots\;q_n]\) satisfies \(Q^T Q = I\)
• Coordinates are obtained by dot products with basis vectors

Common gotchas

• Orthonormality depends on the chosen inner product.
• Numerical orthogonality can degrade with floating point.

Example

In \(\mathbb{R}^2\), the standard basis \((1,0),(0,1)\) is orthonormal; \((3,4)=3(1,0)+4(0,1)\).

How to Compute (Pseudocode)

Input: linearly independent vectors v_1, ..., v_k
Output: orthonormal basis q_1, ..., q_k for the same span

apply Gram-Schmidt (or QR factorization) to the vectors
normalize each resulting basis vector to unit norm
return q_1, ..., q_k

Complexity

• Time: Depends on the orthonormalization method; dense Gram-Schmidt/QR workflows are polynomial in vector dimension and number of vectors
• Space: Depends on storing the input vectors and orthonormal basis (and any factorization workspace)
• Assumptions: Numerical stability is method-dependent (Householder QR is usually preferred over classical Gram-Schmidt)

See also

• Projection