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Given the following lemma on rank-$1$ matrices which I think I understand

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How could I deduce the following on the characteristic polynomial of $A_s$? I don't get it just using multilinearity of determinant. The hypothesis of Theorem 1 is just that $A,B \geq 0$

enter image description here

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  • $\begingroup$ Please consider transcribing the screenshot. Where is the screenshot from? $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 10, 2023 at 20:08
  • $\begingroup$ epubs.siam.org/doi/abs/10.1137/130935537 Unfortunately, I can't share full article. I don't know if it can consulted online $\endgroup$
    – jacopoburelli
    Commented Dec 10, 2023 at 20:14
  • $\begingroup$ Please include the full reference and the DOI link in the question itself $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 10, 2023 at 20:17
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    $\begingroup$ Note: the statement $\lambda_2 \gt \rho(A)$ doesn't make sense based on what you've told us. I suggest taking a different route: try to prove the equivalent statement $p_s = p_0 + s\cdot (p_1-p_0)$, use $\det\big(C - tI\big)$ for characteristic polynomial (you can play around with signs later), write $B-A = \mathbf {uv}^T$, work in the ring $\mathbb R[s, t]$ and use the the adjugate form of the of the Matrix Determinant Lemma. $\endgroup$
    – user8675309
    Commented Dec 11, 2023 at 19:29
  • $\begingroup$ The reference suppose that the matrix is invertible, which might not be my case @user8675309 $\endgroup$
    – jacopoburelli
    Commented Dec 11, 2023 at 21:48

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