  For a matrix ${\bf A}$ with linearly independent columns, this work studies
to use its normalization $\bar{\bf A}$ and ${\bf A}$ itself to approximate its
orthonormalization $\bf V$. We theoretically analyze the order of the
approximation errors as $\bf A$ and $\bar{\bf A}$ approach ${\bf V}$,
respectively. Our conclusion is able to explain the fact that a high
dimensional Gaussian matrix can well approximate the corresponding truncated
Haar matrix. For applications, this work can serve as a foundation of a wide
variety of problems in signal processing such as compressed subspace
clustering.
