Orthogonal Matrix - Rectangular Matrices

Rectangular Matrices

If Q is not a square matrix, then the conditions QTQ = I and QQT = I are not equivalent. The condition QTQ = I says that the columns of Q are orthonormal. This can only happen if Q is an m×n matrix with n ≤ m. Similarly, QQT = I says that the rows of Q are orthonormal, which requires n ≥ m.

There is no standard terminology for these matrices. They are sometimes called "orthonormal matrices", sometimes "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".