Inverting Product of Vectors?

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Let $j$ be an $n*1$ vector. Then, $jj^T$ is an $n*n$ singular matrix. Is the pseudo-inverse or any other application of Singular Value Decomposition going to lead to a meaningful result in this case? For example, in equation $$jj^Tb=aj$$ where $j$ and $b$ are $n*1$ vectors and $a$ is a scalar, is there a way of getting a meaningful solution for vector $b$?