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Given a diagonal matrix $X$, what is the best way to factor it using two vectors as $X=u \cdot v^T$? How do we find such vectors $u$ and $v$ for a diagonal matrix $X$?

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    @pvep: Do you say you *define* a diagonal matrix to be one of the form $u \cdot v^T$? Please don't, every calls a diagonal matrix one with nonzero entries only on the diagonal. Matrices of the form $u \cdot v^T$ are called rank-$1$ matrices.2012-01-25

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If a diagonal matrix $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ can we written as $u\cdot v^T$, with $u=(u_1,\ldots,u_n)^T$ and $v=(v_1,\ldots,v_n)^T$, and if $\lambda_i\neq 0$ and $\lambda_k\neq 0$ for two distinct $i,k\in\{1,\ldots,n\}$, then $u_iv_i\neq 0$ and $u_kv_k\neq 0$. So $u_i,v_i,u_k,v_k$ are different from $0$, and so is for example $u_iv_k$. In particular $u\cdot v^T$ is not diagonal.