An Orthonormal Vector Set is a Set of Vectors that are Mutually Orthogonal to each other, with each Vector having a Length of 1.
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).
Orthogonal Matrices are denoted by letter \(Q\). Following are some examples of Orthogonal Matrices
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 \(Q^TQ=U^{\dagger}U=I\)).
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