How to compute the inverse of a rank-$1$ matrix

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This is impossible in general: it would mean the associated linear map $\;r\colon\mathbf C^m \longrightarrow \mathbf C^m\; $ is surjective. As the rank of the matrix is $1$, ie. the image of $r$ has dimension $1$, this is possible only if $m=1$.

Another way of proving it is to observe that $$\operatorname{rank}(RX)\le \operatorname{rank}R \le 1.$$

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Updated on April 01, 2020

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  • user350491
    user350491 over 3 years

    I have a rank-1 matrix $R \in \mathcal{C}^{m \times m}$, how to compute another matrix X, such that $RX=I$, where $I$ is an identity matrix.

    • Mathematician 42
      Mathematician 42 over 7 years
      It is impossible unless $m=1$.
    • Mathematician 42
      Mathematician 42 over 7 years
      Why do you ask this question? Surely if you know what rank is you should see that this question makes no sense whatsoever.
    • user350491
      user350491 over 7 years
      Because I have an optimization problem, and the constraint is shown like $\mathbf{S}^{-1}\mathbf{R_{xx}}-\mathbf{S}^{-1}\left(\mathbf{S}^{-1}+a^{-1}\text{diag}(\mathbf{p}) \right)^{-1}\mathbf{S}^{-1}\mathbf{R_{xx}}\succeq \mathbf{Z}$, where $p$ is the only unknown vector. And I want to write it into a Linear Matrix Inequality, such that CVX SDP can be used to solve my problem.
    • Mathematician 42
      Mathematician 42 over 7 years
      If you want us to understand what exactly this means, you need to be more specific. What are $S, R_{xx},x,a $ and $Z$? What is $diag(\cdot)$ of a vector and what is the meaning of the inequality sign?
    • user350491
      user350491 over 7 years
      $S,a,Z$ are known already, $R_{xx}$ is the matrix as $R$ I used in my original problem statement. $diag(p)$ means a diagonal matrix, which is composed by $p$ vector. $X \succeq Y$ means $X-Y$ is positive semidefinite.
    • Mathematician 42
      Mathematician 42 over 7 years
      $S,a,Z$ are matrices? Of what sizes? $R_{xx}$ is a rank one matrix? Of what size? Why the subscript? $diag(p)$ is any diagonal matrix multiplied by $p$? You really need to be more specific, we can't simply guess what you mean.
    • Ben Grossmann
      Ben Grossmann over 7 years
      @user350491 perhaps you should ask a new question about your SDP
    • user350491
      user350491 over 7 years
      $a$ is a known constant. $S,Z R_{xx} \in \mathcal{C}^{m\times m}$, and $R_{xx}$ is rank-1, and $S,Z$ can be full rank. $diag(p) \in \mathcal{R}^{m\times m}$ is a diagonal matrix composed by $p \in \mathcal{R}^{m\time 1}$, which means that $p$ is specified as the diagonal elements of $diag{p}$. So my question is how to find a matrix $X$, such that $R_{xx}X=I$, how to consider my constraint in CVX SDP?