# The pseudoinverse of a rank one matrix

Let $$A$$ be a $$m\times n$$ matrix with real coefficients. It is proved, for example here Derive the Pseudo Inverse (Moore Penrose) of Rank 1 Matrix as a Scalar Multiple of Its Transpose that if $$A$$ is a rank one matrix, then the pseudoinverse $$A^+$$ is $$\frac{1}{c}A^*$$, where $$c$$ is the sum of the squares of the entries of $$A$$. Is the converse true?

Yes. $$A A^+$$ is the orthogonal projection on the range $$\text{Ran}(A)$$. Its trace is the rank of $$A$$. But $$\text{tr}(A A^*)$$ is the sum of the squares of the entries of $$A$$. Thus if your condition holds, $$\text{rank}(A) =\text{tr}(A A^+)= \text{tr}(A A^*/c) = c/c = 1$$