I am wondering if there is a way to formulate or generate a matrix $X \in R^{n\times n}$ whose column vectors $\{x\_1,\dotsc,x\_i,\dotsc,x\_n\}$ are such that $x\_i$ and $x\_j$ are orthogonal iff $\lvert i-j\rvert>m$. The matrix $X^T X$ would be a band matrix with zeros off-band. I do not have a particular requirement for $x\_i \cdot x\_j$ when $\lvert i-j\rvert\le m$ other than being non-zero. I do require, however, that all vectors are distinct.


One example of $X$ would be a band matrix with bandwidth $2m+1$ and zero off-band. However, I am looking for a more general formula. I would like to somehow link it to SVD, but I am not sure how to. Any ideas would be appreciated!



Generate an ordered set of mostly orthogonal vectors $\{x_i\}$ where $x_i \cdot x_j =0$ iff $\lvert i-j\rvert >m$
It seems that the band matrix example is an essentially sufficient characterization, up to a choice of basis.


The orthogonality constraint forces zeroes in the QR decomposition $X=QR$: $R$ is an upper triangular $m$-band matrix (and $Q$ is our usual orthogonal matrix.)


Conversely, any such pair $Q$, $R$ generates an $X$ where any two column vectors further than $m$ apart are constructed from different column vectors in $Q$ and are thus orthogonal.


