Matrix - Orthonormal

An orthonormal matrix $U$ is a square matrix which satisfies:

$$U^{T}U=U{U}^T=I$$

where:

• $U^T$ is the transpose of $U$
• $I$ is the identity matrix

Other Definition

An orthonormal matrix is a square matrix whose columns and rows form a set of orthonormal vectors.

Meaning

An orthonormal matrix represents a linear transformation that preserves the length of vectors and the angles between them. This means that when you apply an orthonormal matrix to a vector, the resulting vector has the same magnitude as the original vector.

Properties

• The columns (and rows) of an orthonormal matrix are orthogonal set of unit vectors.
• The inverse of an orthonormal matrix is equal to its transpose:

$$U^{-1}=U^T$$

• Orthonormal matrices preserve the dot product of vectors:

$$(Ux)\cdot (Uy)=x\cdot y$$

• The determinant of an orthonormal matrix is either +1 or -1.

Example

$$U=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$$