Orthonormal Matrix

Definition

${\mathbf{Q}}^{⊺}\mathbf{Q}=\mathbf{Q}{\mathbf{Q}}^{⊺}=\mathbf{I}\text{and}{\mathbf{Q}}^{⊺}={\mathbf{Q}}^{-1}$ The matrix whose column and rows are orthonormal vectors.

Examples

• Rotation Matrix
• Permutation Matrix

Facts

Every orthonormal matrix is Unitary Matrix.

For a vector $\mathbf{y}=x_1{\mathbf{q}}_1+x_2{\mathbf{q}}_2+\cdots +x_n{\mathbf{q}}_n$, which can be expressed as a linear combination of column vectors of the orthonormal matrix $\mathbf{Q}$, we can easily find the coefficients $x_i={\mathbf{q}}_i^{⊺}\mathbf{y}$ of the linear combination using the Inner product. $\mathbf{y}=\mathbf{Qx}\to {\mathbf{Q}}^{⊺}\mathbf{y}=\underset{\mathbf{I}}{\underbrace{{\mathbf{Q}}^{⊺}\mathbf{Q}}}\mathbf{x}\to {\mathbf{Q}}^{⊺}\mathbf{y}=\mathbf{x}$