Real Matrices with Orthonormal Columns

Matrix notation

Let $u_1 \dots , u_k \in \R ^n$. Define $U \in \R ^{n \times k}$ so that

\[ U = \bmat{u_1 & u_2 & \cdots & u_k}. \]

Then $\set{u_1, \dots , u_k}$ is an orthonormal set mean

\[ U^\top U = I_k \]

Notice that if $k < n$, $UU^\top \neq I$, since its rank is at most $k$.