What is the householder reflector's singular value? [closed]












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A householder reflector is $I-2ww^*$ where $ w$ is an unit vector.










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closed as off-topic by Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos Dec 17 '18 at 17:19


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    0












    $begingroup$


    A householder reflector is $I-2ww^*$ where $ w$ is an unit vector.










    share|cite|improve this question











    $endgroup$



    closed as off-topic by Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos Dec 17 '18 at 17:19


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos

    If this question can be reworded to fit the rules in the help center, please edit the question.



















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      $begingroup$


      A householder reflector is $I-2ww^*$ where $ w$ is an unit vector.










      share|cite|improve this question











      $endgroup$




      A householder reflector is $I-2ww^*$ where $ w$ is an unit vector.







      linear-algebra






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      edited Dec 17 '18 at 13:23







      Kristy

















      asked Dec 17 '18 at 13:09









      KristyKristy

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      closed as off-topic by Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos Dec 17 '18 at 17:19


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos Dec 17 '18 at 17:19


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Arnaud D., GNUSupporter 8964民主女神 地下教會, Adrian Keister, Paul Frost, José Carlos Santos

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
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          1












          $begingroup$

          Since $H = I- 2ww^*$ is hemitian, the singular values of $H$are just the absolute values of the eigenvalues of $H$.



          Notice that if $uperp w$, then $$Hu = u - 2ww^*u = u-2langle u,wrangle w = u$$



          so $u$ is an eigenvector for $H$ with eigenvalue $1$. If you are working in $mathbb{C}^n$, you can pick $n-1$ linearly independent vectors orthogonal to $v$ so the multiplicity of the eigenvalue $1$ is at least $n-1$.



          On the other hand, we have
          $$Hw = w - 2ww^*w = w - 2|w|^2w = w-2w = -w$$



          so $w$ is the eigenvector for $H$ with eigenvalue $-1$. Therefore, the only eigenvalues are $pm 1$ so the desired singular value is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
            $endgroup$
            – Kristy
            Dec 17 '18 at 13:48












          • $begingroup$
            @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
            $endgroup$
            – mechanodroid
            Dec 17 '18 at 13:56




















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          Since $H = I- 2ww^*$ is hemitian, the singular values of $H$are just the absolute values of the eigenvalues of $H$.



          Notice that if $uperp w$, then $$Hu = u - 2ww^*u = u-2langle u,wrangle w = u$$



          so $u$ is an eigenvector for $H$ with eigenvalue $1$. If you are working in $mathbb{C}^n$, you can pick $n-1$ linearly independent vectors orthogonal to $v$ so the multiplicity of the eigenvalue $1$ is at least $n-1$.



          On the other hand, we have
          $$Hw = w - 2ww^*w = w - 2|w|^2w = w-2w = -w$$



          so $w$ is the eigenvector for $H$ with eigenvalue $-1$. Therefore, the only eigenvalues are $pm 1$ so the desired singular value is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
            $endgroup$
            – Kristy
            Dec 17 '18 at 13:48












          • $begingroup$
            @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
            $endgroup$
            – mechanodroid
            Dec 17 '18 at 13:56


















          1












          $begingroup$

          Since $H = I- 2ww^*$ is hemitian, the singular values of $H$are just the absolute values of the eigenvalues of $H$.



          Notice that if $uperp w$, then $$Hu = u - 2ww^*u = u-2langle u,wrangle w = u$$



          so $u$ is an eigenvector for $H$ with eigenvalue $1$. If you are working in $mathbb{C}^n$, you can pick $n-1$ linearly independent vectors orthogonal to $v$ so the multiplicity of the eigenvalue $1$ is at least $n-1$.



          On the other hand, we have
          $$Hw = w - 2ww^*w = w - 2|w|^2w = w-2w = -w$$



          so $w$ is the eigenvector for $H$ with eigenvalue $-1$. Therefore, the only eigenvalues are $pm 1$ so the desired singular value is $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
            $endgroup$
            – Kristy
            Dec 17 '18 at 13:48












          • $begingroup$
            @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
            $endgroup$
            – mechanodroid
            Dec 17 '18 at 13:56
















          1












          1








          1





          $begingroup$

          Since $H = I- 2ww^*$ is hemitian, the singular values of $H$are just the absolute values of the eigenvalues of $H$.



          Notice that if $uperp w$, then $$Hu = u - 2ww^*u = u-2langle u,wrangle w = u$$



          so $u$ is an eigenvector for $H$ with eigenvalue $1$. If you are working in $mathbb{C}^n$, you can pick $n-1$ linearly independent vectors orthogonal to $v$ so the multiplicity of the eigenvalue $1$ is at least $n-1$.



          On the other hand, we have
          $$Hw = w - 2ww^*w = w - 2|w|^2w = w-2w = -w$$



          so $w$ is the eigenvector for $H$ with eigenvalue $-1$. Therefore, the only eigenvalues are $pm 1$ so the desired singular value is $1$.






          share|cite|improve this answer









          $endgroup$



          Since $H = I- 2ww^*$ is hemitian, the singular values of $H$are just the absolute values of the eigenvalues of $H$.



          Notice that if $uperp w$, then $$Hu = u - 2ww^*u = u-2langle u,wrangle w = u$$



          so $u$ is an eigenvector for $H$ with eigenvalue $1$. If you are working in $mathbb{C}^n$, you can pick $n-1$ linearly independent vectors orthogonal to $v$ so the multiplicity of the eigenvalue $1$ is at least $n-1$.



          On the other hand, we have
          $$Hw = w - 2ww^*w = w - 2|w|^2w = w-2w = -w$$



          so $w$ is the eigenvector for $H$ with eigenvalue $-1$. Therefore, the only eigenvalues are $pm 1$ so the desired singular value is $1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 17 '18 at 13:31









          mechanodroidmechanodroid

          27.8k62447




          27.8k62447












          • $begingroup$
            If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
            $endgroup$
            – Kristy
            Dec 17 '18 at 13:48












          • $begingroup$
            @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
            $endgroup$
            – mechanodroid
            Dec 17 '18 at 13:56




















          • $begingroup$
            If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
            $endgroup$
            – Kristy
            Dec 17 '18 at 13:48












          • $begingroup$
            @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
            $endgroup$
            – mechanodroid
            Dec 17 '18 at 13:56


















          $begingroup$
          If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
          $endgroup$
          – Kristy
          Dec 17 '18 at 13:48






          $begingroup$
          If I calculate the eigenvalue like this:$|H-lambda I|=|(1-lambda )I-2ww^*|=begin{vmatrix} 1-lambda & w^*\ 2w & I end{vmatrix}=begin{vmatrix} 1-lambda+2w^*w & w^*\ 0 & I end{vmatrix}=0$,implying that $lambda=3$, where did I go wrong?
          $endgroup$
          – Kristy
          Dec 17 '18 at 13:48














          $begingroup$
          @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
          $endgroup$
          – mechanodroid
          Dec 17 '18 at 13:56






          $begingroup$
          @Kristy How did you get $$det((1-lambda) I - 2ww^*) = begin{vmatrix} 1-lambda & w^* \ 2w & Iend{vmatrix}$$ It doesn't seem correct.
          $endgroup$
          – mechanodroid
          Dec 17 '18 at 13:56





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