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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.