[Math] The Columns of Semi Orthogonal Matrix

Question

Given a semi-orthogonal matrix whose dimension is $m \times n$, where $m \le n$, $a$ and $b$ are two columns of the matrix selected randomly. Is the proposition "$\langle a, b\rangle = 0 \text{ or } 1$" right?

Thanks.

Answer 1

Take a semi-orthogonal matrix $1 / 2 \begin{bmatrix} 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \end{bmatrix}$ as an example. The proposition is false.

In fact since we can supply some rows to make a semi-orthogonal matrix to be a square orthogonal matrix, we can see that the inner-product of a column of a semi-orthogonal matrix with itself is no greater than the inner-product of the corresponding column of the corresponding orthogonal matrix, which equal 1. The equality only holds when the elements you supplied in that column are all 0.