39. Orthogonal and Orthonormal Sets¹

Dated: 18-06-2025

Orthogonal Set¹

The set¹ of non zero vectors² \(S = \{\vec u_1, \vec u_2, \ldots, \vec u_p\} \in \mathbb R^n\) is called the orthogonal set if each vector² \(\vec u \in S\) is mutually orthogonal.

\[\vec o \notin S \ \land \ \vec u_i \cdot \vec u_j = o \ \forall i \ne j \ \land i, j \in [1, p] \]

Theorem

Let \(S\) be the orthogonal set and \(\mathbb W = \text{Span}(S)\) then \(S\) is linearly independent³ and a basis for \(\mathbb W\).

Orthogonal Basis

If \(\mathbb W = \text{Span}(S)\) where \(S\) is an orthogonal set then \(S\) is also called the orthogonal basis.

Theorem

If \(S\) is orthogonal basis for \(\mathbb W \in \mathbb R^n\) then

\[\vec y = c_1 \vec u_1 + c_2 \vec u_2 + \cdots c_p \vec u_p \quad \text{where } \vec y \in \mathbb W\]

An Orthogonal Projection (Decomposition of a Vector into the Sum of Two Vectors

\[y = y^\prime + z\]

where \(y^\prime = c \vec u\) and \(\vec z \perp \vec u\).

Orthonormal Set

\(S\) is called an orthonormal set if it is an orthogonal set of unit vectors.²

Orthonormal Basis

\(S\) is called orthonormal basis for \(\mathbb W \in \mathbb R^n\) if it spans \(\mathbb W\) and is also an orthonormal set.

Theorem

A \(m \times n\) matrix⁴ \(U\) has orthonormal columns if and only if \(U^T U = I\).

Theorem

Let \(\textbf U_{m \times n}\) be a matrix⁴ with orthonormal columns and let \(\vec x, \vec y \in \mathbb R^n\) then

• \[||U \vec x|| = ||\vec x||\]
• \[(U \vec x) \cdot (U \vec y) = \vec x \cdot \vec y\]
• \[(U \vec x) \cdot (U \vec y) = 0 \iff \vec x \cdot \vec y = 0\]

References

Read more about notations and symbols.

---

1. Read more about sets. ↩↩↩

2. Read more about vectors. ↩↩↩

3. Read more about linear dependency. ↩

4. Read more about matrices. ↩↩