Polar Decomposition of non-square matrix on an Econometrics problem

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I have the following problem at hand on my Econometrics problem set. I am stuck at item a), which is basically a linear algebra problem:




Consider the population model $y=Xbeta +u$,
where $X$ is an $ntimes k $ matrix and $u|Xsim Nleft( 0,sigma ^2I_nright) $.



a) Give the polar decomposition of the matrix $X$.



b) Use (a) to show that $y^prime My/sigma ^2$ has a chi-square $n-k$
distribution, where $M = I_n - X(X'X)^-1X'$.




I was able to prove the second result but I am stuck at the first. Is there a neat way to represent the polar decomposition in this setup? I have no intuition. Thanks in advance!




linear-algebra statistics normal-distribution




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    up vote
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    I have the following problem at hand on my Econometrics problem set. I am stuck at item a), which is basically a linear algebra problem:




    Consider the population model $y=Xbeta +u$,
    where $X$ is an $ntimes k $ matrix and $u|Xsim Nleft( 0,sigma ^2I_nright) $.



    a) Give the polar decomposition of the matrix $X$.



    b) Use (a) to show that $y^prime My/sigma ^2$ has a chi-square $n-k$
    distribution, where $M = I_n - X(X'X)^-1X'$.




    I was able to prove the second result but I am stuck at the first. Is there a neat way to represent the polar decomposition in this setup? I have no intuition. Thanks in advance!




    linear-algebra statistics normal-distribution




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have the following problem at hand on my Econometrics problem set. I am stuck at item a), which is basically a linear algebra problem:




      Consider the population model $y=Xbeta +u$,
      where $X$ is an $ntimes k $ matrix and $u|Xsim Nleft( 0,sigma ^2I_nright) $.



      a) Give the polar decomposition of the matrix $X$.



      b) Use (a) to show that $y^prime My/sigma ^2$ has a chi-square $n-k$
      distribution, where $M = I_n - X(X'X)^-1X'$.




      I was able to prove the second result but I am stuck at the first. Is there a neat way to represent the polar decomposition in this setup? I have no intuition. Thanks in advance!




      linear-algebra statistics normal-distribution




      share|cite|improve this question











      I have the following problem at hand on my Econometrics problem set. I am stuck at item a), which is basically a linear algebra problem:




      Consider the population model $y=Xbeta +u$,
      where $X$ is an $ntimes k $ matrix and $u|Xsim Nleft( 0,sigma ^2I_nright) $.



      a) Give the polar decomposition of the matrix $X$.



      b) Use (a) to show that $y^prime My/sigma ^2$ has a chi-square $n-k$
      distribution, where $M = I_n - X(X'X)^-1X'$.




      I was able to prove the second result but I am stuck at the first. Is there a neat way to represent the polar decomposition in this setup? I have no intuition. Thanks in advance!





      linear-algebra statistics normal-distribution





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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      Raul Guarini

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