Could anyone help me solve this equation for $c$:
$$a^T(B + sc^T) = \vec{0}.$$
Where$a$ is an $i \times 1$ vector, $B$ is an $i \times v$ matrix, $s$ is an $i \times 1$ vector of $1$:s i.e.$$s=\begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}$$$c$ is an $v \times 1$ vector and the $0$ vector is $1 \times v$.
My initial approach was to do:
\begin{align*} a^TB + a^T(sc^T) &= \vec{0} \\ a^T(sc^T) &= -a^TB \end{align*}
but not sure where to go from there. Thanks in advance.
$\endgroup$1 Answer
$\begingroup$By associativity of matrix multiplication, you can get$$(a^\top s)c^\top = -a^\top B,$$where $a^\top s$ is the scalar (well, $1 \times 1$ matrix) $a \cdot s$. Provided that $a$ and $s$ are not orthogonal, we can get$$c^\top = \frac{-a^\top B}{a^\top s} \implies c = \frac{-B^\top a}{a^\top s}.$$If $a$ and $s$ are orthogonal, then we can't solve for $c$; either $a^\top B = 0$, in which case any $c$ will work, or $a^\top B \neq 0$, in which case no $c$ will work.
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