Orthonormal

Orthonormal Vectors

A set of $n$ column vectors is orthonormal iff:

• They are pairwise orthogonal AND
• $\forall i \in \{1,\dots,n\},\; \|v_i\| = 1$

Orthonormal Matrix

A matrix $Q$ whose columns are orthonormal. For real matrices: $Q^T Q = I$ (orthogonal matrix). For complex matrices: $Q^* Q = I$ (unitary matrix).

Key property: $Q^{-1} = Q^T$ (real) or $Q^{-1} = Q^*$ (complex).

Why Orthonormal Bases

• Preserve lengths and angles under transformation: $\|Qx\| = \|x\|$.
• Coordinates in an orthonormal basis are computed by a simple dot product, no matrix inversion needed.
• Numerically stable: no accumulation of rounding errors across operations.