Constraints to make a matrix triangular

Question

I have a rank one matrix given below

$$A = bc^T$$

where $A \in R^{n \times n}$, $b \in R^n$ and $c \in R^n$. I have where the matrix is made of rank components where each component should be a lower triangular matrix. How should I put constraints on $b$ and $c$ such that $A$ is a lower triangular matrix? and how do I solve them?

Answer 1

Let $j$ be the index (if exists) such that $c_j\ne0=c_{j+1}=\cdots=c_n$. That is, let $c_j$ be the last nonzero entry of $c$.

If $A=bc^T$ is lower triangular, we must have $a_{1j}=a_{2j}=\cdots=a_{j-1,j}=0$. This imposes some necessary conditions on the entries of $b$. One may verify that these conditions are also sufficient for $bc^T$ to be lower triangular.

Answer 2

Hint: $A_{ij} = b_i c_j$ and you want $A_{ij} = 0$ whenever $j > i$.

Hint: alternatively, note that the columns of $A$ are multiples of $b$. How can this be lower triangular?

I think you need either $b_1 = \cdots = b_{n-1} = 0$ or $c_2 = \cdots = c_n = 0$.

Answer 3

We have $$a_{ij}=b_ic_j$$therefore $$\text{if }b_i\ne 0 \text{ then }c_j=0,j>i$$the algorithm then goes like this: for any vector $b$, find $i$ as the maximum index for which $b_i=0$. Then any vector $c$ for which $c_j=0$ for all $j>i$ is qualified to make an upper-triangular matrix in companion to $b$ in form of $bc^T$.