Suppose that $X$ is a $p \times n$ matrix and that the rank of $X$ is $p$. We may then decompose $X$ as:
$$ X = LCR^T $$
where we have $L$ is a $p\times p$ matrix, $C$ a diagonal $p\times p$ matrix, and $R^T$ a $p\times n$ matrix where $p \leq n$.
Let us then define $M(a,b) = l_ar_b^T$ as an outer product for $a = 1, \ldots, p$ and $b = 1, \ldots, p$. Here, $l_a$ and $r_b$ are vectors in $L,R$ above.
I would like to show that the vectors $M(a,b) = l_av_b^T$ are of unit length and are also mutually orthogonal in the Matrix Norm sense. Additionally, how could I find the coordinates of $X$ in terms of my set of vectors?
My approach is to use a Frobenius norm, but it is going nowhere. Is there something I'm missing here? Thanks.