Orthogonal and Unitary Matrices

1. An Orthonormal Vector Set is a Set of Vectors that are Mutually Orthogonal to each other, with each Vector having a Length of 1.
2. A Matrix comprising of only Orthonormal Vector Set as its Columns is called an Orthogonal Matrix (or a Unitary Matrix if the any of the Vectors have Complex Components).
3. Orthogonal Matrices are denoted by letter \(Q\). Following are some examples of Orthogonal Matrices
   
   \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}\hspace{5mm}\begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}\hspace{5mm} \begin{bmatrix} \frac{3}{\sqrt{13}} & \frac{2}{\sqrt{13}} \\ \frac{-2}{\sqrt{13}} & \frac{3}{\sqrt{13}}\end{bmatrix} \hspace{5mm} \begin{bmatrix} \frac{2}{3} & \frac{-1}{\sqrt{18}} \\ \frac{2}{3} & \frac{-1}{\sqrt{18}} \\ \frac{1}{3} & \frac{4}{\sqrt{18}}\end{bmatrix}\)
   
   Unitary Matrices are denoted by letter \(U\) (or sometimes by letter \(Q\) also). Following are some examples of Unitary Matrices
   
   \(\begin{bmatrix}0.62361+0.26726i & 0.39321+0.62054i \\ 0.7127-0.17817i & -0.6359-0.23654i\end{bmatrix} \hspace{.5cm} \begin{bmatrix}0.26621-0.44368i & 0.37561-0.23723i \\ 0.70989 & 0.13399+0.24162i \\ -0.17747-0.44368i & -0.30093+0.79734i \end{bmatrix}\)
4. The Dual of any Orthogonal Matrix \(Q\) (or Unitary Matrix \(U\)) is the Matrix itself. Since for any Matrix \(A\) with Linearly Independent Vectors \({A_D}^TA=A^T{A_D}=I\) (or if \(A\) is Complex Matrix then \({A_D}^{\dagger}A=A^{\dagger}{A_D}=I\)), therefore the Matrices \(Q^TQ\) and \(U^{\dagger}U\) are Identity Matrices (i.e \(Q^TQ=U^{\dagger}U=I\)).
5. The Transpose of any Orthogonal Square Matrix \(Q\) (or Conjugate Transpose of any Unitary Square Matrix \(U\)) is same as its Inverse. That is, for all Square Orthogonal and Unitary Matrices
   
   \(Q^{-1}=Q^T \hspace{5mm} \Rightarrow Q^{-1}Q=Q^TQ=QQ^{-1}=QQ^T=I\)
   
   \(U^{-1}=U^\dagger \hspace{5mm} \Rightarrow U^{-1}U=U^{\dagger}U=UU^{-1}=UU^{\dagger}=I\)
   
   The Identity Matrices, Permutation Matrices, Rotation Matrices and Reflection Matrices are some examples of Orthogonal Square Matrices.
6. The Determinant of Any Orthogonal Square Matrix is \(\pm\hspace{1mm}1\). The Absolute Value of Determinant of Any Unitary Square Matrix is \(1\).
7. Any Real Orthogonal Square Matrix of Order 2 or 3 having a Determinant Value of \(1\) is a Rotation Matrix.
8. Any Real Symmetric Orthogonal Square Matrix of Any Order having a Determinant Value of \(\pm\hspace{1mm}1\) is a Reflection Matrix.
9. Any Real Non-Symmetric Orthogonal Square Matrix of Order 3 having a Determinant Value of \(-1\) is a Roto-Reflection Matrix / Improper Rotation Matrix.
10. Pre-Multiplying any Vector with any Orthogonal/Unitary Matrix does not change the Length/Magnitude of the Vector.