Matrix representation as product of two matrix

Question

I have a $m \times m$ matrix and I want to represent it as a product of a $m \times 1$ & $1 \times m$ matrix. Could some one please give me the steps to do this?

Answer 1

If the rank of the matrix is one then the rows are linearly dependent. So, if this is the case, we can find scalars $\alpha_i$ such that $r_i$ = $\alpha_i r_1$ (where $r_i$ denotes row $i$) then your original matrix is

$$A = \left(\begin{matrix} 1 \\ \alpha_2 \\ \alpha_3 \\ \vdots \\ \alpha_m\end{matrix}\right) \left(\begin{matrix} r_1^1 & \dots & r_1^m \end{matrix}\right)$$

where $r_1 = \left(\begin{matrix} r_1^1 & \dots & r_1^m \end{matrix}\right)$. Moreover, if a matrix is given as a product of $a\in \mathbb{R}^n, b\in \mathbb{R}^m$ in the sense $A = a b^{T} \in \mathbb{R}^{n\times m}$ then $A$ has rank $\leqq 1$ because the (possibly zero) rows are linearly dependent. If the rank of your matrix is greater than $1$ then such a decomposition does not exist.