Proof of a theorem in distribution theory (probably Cochran's Theorem)

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I want to prove the following theorem (I'm not sure but probably it is called Cochran's Theorem) :



Let $X$ follows $n$-variate multivariate normal distribution with mean vector $mathbfmu$ and covariance matrix $Sigma$. Also let $A_1,A_2,dots,A_m$ be symmetric projection matrices of order $n$ such that $rank(A_i)=r_i$ and $sum_i=1^m A_i=I$. Let $U_i=X^TA_iX$. Show that $U_i$'s are independent random variables, respectively distributed as $chi_r_i^2$.



I cannot find the idea to attack the problem. If anyone can give me any lead (even for the simple $m=2$ case, which I may generalize), I'll me much grateful. Thank you.






linear-algebra probability-distributions normal-distribution independence chi-squared







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  • en.wikipedia.org/wiki/Cochran%27s_theorem
    – Hugo
    Aug 30 at 6:51










  • I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
    – Eugenia
    Aug 30 at 9:58














up vote
0
down vote

favorite












I want to prove the following theorem (I'm not sure but probably it is called Cochran's Theorem) :



Let $X$ follows $n$-variate multivariate normal distribution with mean vector $mathbfmu$ and covariance matrix $Sigma$. Also let $A_1,A_2,dots,A_m$ be symmetric projection matrices of order $n$ such that $rank(A_i)=r_i$ and $sum_i=1^m A_i=I$. Let $U_i=X^TA_iX$. Show that $U_i$'s are independent random variables, respectively distributed as $chi_r_i^2$.



I cannot find the idea to attack the problem. If anyone can give me any lead (even for the simple $m=2$ case, which I may generalize), I'll me much grateful. Thank you.






linear-algebra probability-distributions normal-distribution independence chi-squared







share|cite|improve this question





















  • en.wikipedia.org/wiki/Cochran%27s_theorem
    – Hugo
    Aug 30 at 6:51










  • I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
    – Eugenia
    Aug 30 at 9:58












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I want to prove the following theorem (I'm not sure but probably it is called Cochran's Theorem) :



Let $X$ follows $n$-variate multivariate normal distribution with mean vector $mathbfmu$ and covariance matrix $Sigma$. Also let $A_1,A_2,dots,A_m$ be symmetric projection matrices of order $n$ such that $rank(A_i)=r_i$ and $sum_i=1^m A_i=I$. Let $U_i=X^TA_iX$. Show that $U_i$'s are independent random variables, respectively distributed as $chi_r_i^2$.



I cannot find the idea to attack the problem. If anyone can give me any lead (even for the simple $m=2$ case, which I may generalize), I'll me much grateful. Thank you.






linear-algebra probability-distributions normal-distribution independence chi-squared







share|cite|improve this question













I want to prove the following theorem (I'm not sure but probably it is called Cochran's Theorem) :



Let $X$ follows $n$-variate multivariate normal distribution with mean vector $mathbfmu$ and covariance matrix $Sigma$. Also let $A_1,A_2,dots,A_m$ be symmetric projection matrices of order $n$ such that $rank(A_i)=r_i$ and $sum_i=1^m A_i=I$. Let $U_i=X^TA_iX$. Show that $U_i$'s are independent random variables, respectively distributed as $chi_r_i^2$.



I cannot find the idea to attack the problem. If anyone can give me any lead (even for the simple $m=2$ case, which I may generalize), I'll me much grateful. Thank you.






linear-algebra probability-distributions normal-distribution independence chi-squared




linear-algebra probability-distributions normal-distribution independence chi-squared






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 30 at 6:43









Eugenia

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  • en.wikipedia.org/wiki/Cochran%27s_theorem
    – Hugo
    Aug 30 at 6:51










  • I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
    – Eugenia
    Aug 30 at 9:58
















  • en.wikipedia.org/wiki/Cochran%27s_theorem
    – Hugo
    Aug 30 at 6:51










  • I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
    – Eugenia
    Aug 30 at 9:58















en.wikipedia.org/wiki/Cochran%27s_theorem
– Hugo
Aug 30 at 6:51




en.wikipedia.org/wiki/Cochran%27s_theorem
– Hugo
Aug 30 at 6:51












I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
– Eugenia
Aug 30 at 9:58




I saw the proof in Wiki. But I was wondering if one can give ideas on a simpler proof.
– Eugenia
Aug 30 at 9:58















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