Unitary and Orthogonal Transforms

A square matrix $A=[A_1\;A_2\;\cdots\;A_n]$ (

for the ith column vector of

) is unitary if its inverse is equal to its conjugate transpose, i.e., $A^{-1}=A^{*T}$ . In particular, if a unitary matrix is real

, then $A^{-1}=A^T$ and it is orthogonal. Both the column and row vectors ( $A_i, i=1,\cdots,n$ ) of a unitary or orthogonal matrix are orthogonal (perpendicular to each other) and normalized (of unit length), or orthonormal, i.e., their inner product satisfies:

$\begin{displaymath}(A_i,A_j)=A_i^{*T} A_j=\delta_{i,j}=\left\{ \begin{array}{ll} 1 & \mbox{if } i=j 0 & \mbox{otherwise} \end{array} \right. \end{displaymath}$