What actually it means by the words “define the orthogonal projection in the spectral decomposition”?












0












$begingroup$


Define the orthogonal projection in the spectral decomposition for the $2 times 2$ matrix $ begin{bmatrix} 2 & 1 \ 1 & 2 end{bmatrix}$.



Answer:



I can find the orthogonal projection of the above matrix.



Also I can find the spectral decomposition of the above matrix.



But I do not understand the question's meaning.



What actually it means by the word $ text{define the orthogonal projection in the spectral decomposition}$ ?



Help me






linear-algebra terminology







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    That's more than one word.
    $endgroup$
    – Shaun
    Dec 7 '18 at 22:43










  • $begingroup$
    @what can be the meanings?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 22:47










  • $begingroup$
    I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
    $endgroup$
    – Yadati Kiran
    Dec 7 '18 at 23:09












  • $begingroup$
    @YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 23:11
















0












$begingroup$


Define the orthogonal projection in the spectral decomposition for the $2 times 2$ matrix $ begin{bmatrix} 2 & 1 \ 1 & 2 end{bmatrix}$.



Answer:



I can find the orthogonal projection of the above matrix.



Also I can find the spectral decomposition of the above matrix.



But I do not understand the question's meaning.



What actually it means by the word $ text{define the orthogonal projection in the spectral decomposition}$ ?



Help me






linear-algebra terminology







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    That's more than one word.
    $endgroup$
    – Shaun
    Dec 7 '18 at 22:43










  • $begingroup$
    @what can be the meanings?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 22:47










  • $begingroup$
    I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
    $endgroup$
    – Yadati Kiran
    Dec 7 '18 at 23:09












  • $begingroup$
    @YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 23:11














0












0








0





$begingroup$


Define the orthogonal projection in the spectral decomposition for the $2 times 2$ matrix $ begin{bmatrix} 2 & 1 \ 1 & 2 end{bmatrix}$.



Answer:



I can find the orthogonal projection of the above matrix.



Also I can find the spectral decomposition of the above matrix.



But I do not understand the question's meaning.



What actually it means by the word $ text{define the orthogonal projection in the spectral decomposition}$ ?



Help me






linear-algebra terminology







share|cite|improve this question











$endgroup$




Define the orthogonal projection in the spectral decomposition for the $2 times 2$ matrix $ begin{bmatrix} 2 & 1 \ 1 & 2 end{bmatrix}$.



Answer:



I can find the orthogonal projection of the above matrix.



Also I can find the spectral decomposition of the above matrix.



But I do not understand the question's meaning.



What actually it means by the word $ text{define the orthogonal projection in the spectral decomposition}$ ?



Help me






linear-algebra terminology




linear-algebra terminology






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 '18 at 0:45









Arturo Magidin

262k34586910




262k34586910










asked Dec 7 '18 at 22:40









M. A. SARKARM. A. SARKAR

2,1911619




2,1911619








  • 1




    $begingroup$
    That's more than one word.
    $endgroup$
    – Shaun
    Dec 7 '18 at 22:43










  • $begingroup$
    @what can be the meanings?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 22:47










  • $begingroup$
    I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
    $endgroup$
    – Yadati Kiran
    Dec 7 '18 at 23:09












  • $begingroup$
    @YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 23:11














  • 1




    $begingroup$
    That's more than one word.
    $endgroup$
    – Shaun
    Dec 7 '18 at 22:43










  • $begingroup$
    @what can be the meanings?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 22:47










  • $begingroup$
    I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
    $endgroup$
    – Yadati Kiran
    Dec 7 '18 at 23:09












  • $begingroup$
    @YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
    $endgroup$
    – M. A. SARKAR
    Dec 7 '18 at 23:11








1




1




$begingroup$
That's more than one word.
$endgroup$
– Shaun
Dec 7 '18 at 22:43




$begingroup$
That's more than one word.
$endgroup$
– Shaun
Dec 7 '18 at 22:43












$begingroup$
@what can be the meanings?
$endgroup$
– M. A. SARKAR
Dec 7 '18 at 22:47




$begingroup$
@what can be the meanings?
$endgroup$
– M. A. SARKAR
Dec 7 '18 at 22:47












$begingroup$
I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
$endgroup$
– Yadati Kiran
Dec 7 '18 at 23:09






$begingroup$
I guess the question says " find spectral decomposition of $A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$ where $P_{lambda}$ denotes orthogonal projection onto $textbf{eigenspace}$ $V_{lambda}$ ".
$endgroup$
– Yadati Kiran
Dec 7 '18 at 23:09














$begingroup$
@YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
$endgroup$
– M. A. SARKAR
Dec 7 '18 at 23:11




$begingroup$
@YadatiKiran, write clearly and also mention what do you mean by $V_{lambda}$?
$endgroup$
– M. A. SARKAR
Dec 7 '18 at 23:11










1 Answer
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$begingroup$

I presume the question requires to find orthogonal projections $P_{lambda}$ in the spectral decomposition $$fbox{$A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$} $$ where $P_{lambda}$ denotes the orthogonal projection onto eigenspace $V_{lambda}$.



$text{( I am giving details for the ones who do not know )}$



The eigenvalues and corresponding eigenvectors are $lambda=3:,:lambda=1$ with $displaystyle V_{lambda_3}=begin{bmatrix}1\1end{bmatrix}$ and $displaystyle V_{lambda_1}=begin{bmatrix}1\-1end{bmatrix}$. The projection matrix onto $V_{lambda}$ is given by $$fbox{$P=V_{lambda}(V_{lambda}^TV_{lambda})^{-1}V_{lambda}^T$} $$ We have $$P_{lambda_3}=begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}:text{ and}: P_{lambda_1}=begin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix}$$



The spectral decomposition of $A$ is $$A=3cdot begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}+1cdotbegin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix} $$






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    $begingroup$

    I presume the question requires to find orthogonal projections $P_{lambda}$ in the spectral decomposition $$fbox{$A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$} $$ where $P_{lambda}$ denotes the orthogonal projection onto eigenspace $V_{lambda}$.



    $text{( I am giving details for the ones who do not know )}$



    The eigenvalues and corresponding eigenvectors are $lambda=3:,:lambda=1$ with $displaystyle V_{lambda_3}=begin{bmatrix}1\1end{bmatrix}$ and $displaystyle V_{lambda_1}=begin{bmatrix}1\-1end{bmatrix}$. The projection matrix onto $V_{lambda}$ is given by $$fbox{$P=V_{lambda}(V_{lambda}^TV_{lambda})^{-1}V_{lambda}^T$} $$ We have $$P_{lambda_3}=begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}:text{ and}: P_{lambda_1}=begin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix}$$



    The spectral decomposition of $A$ is $$A=3cdot begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}+1cdotbegin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix} $$






    share|cite|improve this answer











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      1












      $begingroup$

      I presume the question requires to find orthogonal projections $P_{lambda}$ in the spectral decomposition $$fbox{$A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$} $$ where $P_{lambda}$ denotes the orthogonal projection onto eigenspace $V_{lambda}$.



      $text{( I am giving details for the ones who do not know )}$



      The eigenvalues and corresponding eigenvectors are $lambda=3:,:lambda=1$ with $displaystyle V_{lambda_3}=begin{bmatrix}1\1end{bmatrix}$ and $displaystyle V_{lambda_1}=begin{bmatrix}1\-1end{bmatrix}$. The projection matrix onto $V_{lambda}$ is given by $$fbox{$P=V_{lambda}(V_{lambda}^TV_{lambda})^{-1}V_{lambda}^T$} $$ We have $$P_{lambda_3}=begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}:text{ and}: P_{lambda_1}=begin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix}$$



      The spectral decomposition of $A$ is $$A=3cdot begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}+1cdotbegin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix} $$






      share|cite|improve this answer











      $endgroup$
















        1












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        1





        $begingroup$

        I presume the question requires to find orthogonal projections $P_{lambda}$ in the spectral decomposition $$fbox{$A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$} $$ where $P_{lambda}$ denotes the orthogonal projection onto eigenspace $V_{lambda}$.



        $text{( I am giving details for the ones who do not know )}$



        The eigenvalues and corresponding eigenvectors are $lambda=3:,:lambda=1$ with $displaystyle V_{lambda_3}=begin{bmatrix}1\1end{bmatrix}$ and $displaystyle V_{lambda_1}=begin{bmatrix}1\-1end{bmatrix}$. The projection matrix onto $V_{lambda}$ is given by $$fbox{$P=V_{lambda}(V_{lambda}^TV_{lambda})^{-1}V_{lambda}^T$} $$ We have $$P_{lambda_3}=begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}:text{ and}: P_{lambda_1}=begin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix}$$



        The spectral decomposition of $A$ is $$A=3cdot begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}+1cdotbegin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix} $$






        share|cite|improve this answer











        $endgroup$



        I presume the question requires to find orthogonal projections $P_{lambda}$ in the spectral decomposition $$fbox{$A=lambda_1P_{lambda_1}+cdots+lambda_nP_{lambda_n}$} $$ where $P_{lambda}$ denotes the orthogonal projection onto eigenspace $V_{lambda}$.



        $text{( I am giving details for the ones who do not know )}$



        The eigenvalues and corresponding eigenvectors are $lambda=3:,:lambda=1$ with $displaystyle V_{lambda_3}=begin{bmatrix}1\1end{bmatrix}$ and $displaystyle V_{lambda_1}=begin{bmatrix}1\-1end{bmatrix}$. The projection matrix onto $V_{lambda}$ is given by $$fbox{$P=V_{lambda}(V_{lambda}^TV_{lambda})^{-1}V_{lambda}^T$} $$ We have $$P_{lambda_3}=begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}:text{ and}: P_{lambda_1}=begin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix}$$



        The spectral decomposition of $A$ is $$A=3cdot begin{bmatrix}dfrac12 &dfrac12\dfrac12 &dfrac12end{bmatrix}+1cdotbegin{bmatrix}dfrac12 &dfrac{-1}{2}\dfrac{-1}{2}&dfrac12end{bmatrix} $$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 8 '18 at 12:29

























        answered Dec 7 '18 at 23:42









        Yadati KiranYadati Kiran

        1,769619




        1,769619






























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