Prove that the following matrix is semidefinite.

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Let $X=[X_1, X_2]$ be a full column rank matrix, where $X_1$ is a $ntimes p$ matrix and $X_2$ is a $ntimes q$ matrix. Both $X_1$ and $X_2$ are full column rank matrices. That is to say, $rm rank(X)=p+q$, $rm rank(X_1)=p$ and $rm rank(X_2)=q$. Prove that the matrix $X(X^TX)^-1X^T-X_1(X_1^TX_1)^-1X_1^T$ is a positive semidefinite matrix.



I try to use Woodbury inverse formula, but it seems fails. Thanks for your help:)




matrices positive-semidefinite




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  • Are all of those matrices real?
    – Mostafa Ayaz
    Jul 22 at 8:56










  • Thank you Mostafa. Yes, there are real.
    – Ning Zheng
    Jul 22 at 8:57










  • Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
    – Mostafa Ayaz
    Jul 22 at 8:59











  • Yes, of course. Otherwise they are full row rank, instead of full column rank.
    – Ning Zheng
    Jul 22 at 9:08














up vote
0
down vote

favorite












Let $X=[X_1, X_2]$ be a full column rank matrix, where $X_1$ is a $ntimes p$ matrix and $X_2$ is a $ntimes q$ matrix. Both $X_1$ and $X_2$ are full column rank matrices. That is to say, $rm rank(X)=p+q$, $rm rank(X_1)=p$ and $rm rank(X_2)=q$. Prove that the matrix $X(X^TX)^-1X^T-X_1(X_1^TX_1)^-1X_1^T$ is a positive semidefinite matrix.



I try to use Woodbury inverse formula, but it seems fails. Thanks for your help:)




matrices positive-semidefinite




share|cite|improve this question



















  • Are all of those matrices real?
    – Mostafa Ayaz
    Jul 22 at 8:56










  • Thank you Mostafa. Yes, there are real.
    – Ning Zheng
    Jul 22 at 8:57










  • Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
    – Mostafa Ayaz
    Jul 22 at 8:59











  • Yes, of course. Otherwise they are full row rank, instead of full column rank.
    – Ning Zheng
    Jul 22 at 9:08












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $X=[X_1, X_2]$ be a full column rank matrix, where $X_1$ is a $ntimes p$ matrix and $X_2$ is a $ntimes q$ matrix. Both $X_1$ and $X_2$ are full column rank matrices. That is to say, $rm rank(X)=p+q$, $rm rank(X_1)=p$ and $rm rank(X_2)=q$. Prove that the matrix $X(X^TX)^-1X^T-X_1(X_1^TX_1)^-1X_1^T$ is a positive semidefinite matrix.



I try to use Woodbury inverse formula, but it seems fails. Thanks for your help:)




matrices positive-semidefinite




share|cite|improve this question











Let $X=[X_1, X_2]$ be a full column rank matrix, where $X_1$ is a $ntimes p$ matrix and $X_2$ is a $ntimes q$ matrix. Both $X_1$ and $X_2$ are full column rank matrices. That is to say, $rm rank(X)=p+q$, $rm rank(X_1)=p$ and $rm rank(X_2)=q$. Prove that the matrix $X(X^TX)^-1X^T-X_1(X_1^TX_1)^-1X_1^T$ is a positive semidefinite matrix.



I try to use Woodbury inverse formula, but it seems fails. Thanks for your help:)





matrices positive-semidefinite





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 22 at 8:49









Ning Zheng

112




112











  • Are all of those matrices real?
    – Mostafa Ayaz
    Jul 22 at 8:56










  • Thank you Mostafa. Yes, there are real.
    – Ning Zheng
    Jul 22 at 8:57










  • Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
    – Mostafa Ayaz
    Jul 22 at 8:59











  • Yes, of course. Otherwise they are full row rank, instead of full column rank.
    – Ning Zheng
    Jul 22 at 9:08
















  • Are all of those matrices real?
    – Mostafa Ayaz
    Jul 22 at 8:56










  • Thank you Mostafa. Yes, there are real.
    – Ning Zheng
    Jul 22 at 8:57










  • Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
    – Mostafa Ayaz
    Jul 22 at 8:59











  • Yes, of course. Otherwise they are full row rank, instead of full column rank.
    – Ning Zheng
    Jul 22 at 9:08















Are all of those matrices real?
– Mostafa Ayaz
Jul 22 at 8:56




Are all of those matrices real?
– Mostafa Ayaz
Jul 22 at 8:56












Thank you Mostafa. Yes, there are real.
– Ning Zheng
Jul 22 at 8:57




Thank you Mostafa. Yes, there are real.
– Ning Zheng
Jul 22 at 8:57












Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
– Mostafa Ayaz
Jul 22 at 8:59





Also what about $p$ , $q$ and $n$? Is $ple n$ and $qle n$ and $p+qle n$?
– Mostafa Ayaz
Jul 22 at 8:59













Yes, of course. Otherwise they are full row rank, instead of full column rank.
– Ning Zheng
Jul 22 at 9:08




Yes, of course. Otherwise they are full row rank, instead of full column rank.
– Ning Zheng
Jul 22 at 9:08















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