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Say I have a vector ${\bf x} = [x_1 \:\: x_2 \:\: x_3 \dots x_n]$. Consider the matrix ${\bf X}$ where $ij$-th element is the ratio $$\frac{x_i}{x_j}$$ Does this matrix have a special name?

Pardon if this is common knowledge, but a quick google search did not reveal anything.

EDIT: Alternatively if I have two vectors $\bf x$ and $\bf y$ and the $ij$-th element is $$\frac{x_i}{y_j}$$ what then?

kbau
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1 Answers1

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I don't know if this matrix has a special name, but it does have a special form. If you let $y = 1/x$ be the vector of all multiplicative inverses of entries of $x$, then your matrix is the outer-product $xy^T$. In particular, this says your matrix has rank 1. It may also simplify computations involving the matrix.

user2566092
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