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Math Help - Rank-1 Projections

  1. #1
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    Rank-1 Projections

    Show that if u has a unit length, then the rank-1 matrix P=uu^t is a projection matrix, meaning P^2=P and P^t=P. By choosing u = a / ||a||, P becomes the projection onto the line through a, and Pb is the point p = xa. Rank-1 projections correspond exactly to a least squares problem in 1 unknown.

    OK... I don't really know what they're talking about... The only relevant equation I have is P=A(A^tA)^-1A^t. I don't know how to apply it. Please help!
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  2. #2
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    Anyone? I know this question is not that difficult, but I think I'm missing something...
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    Show that if u has a unit length, then the rank-1 matrix P=uu^t is a projection matrix, meaning P^2=P and P^t=P. By choosing u = a / ||a||, P becomes the projection onto the line through a, and Pb is the point p = xa. Rank-1 projections correspond exactly to a least squares problem in 1 unknown.

    OK... I don't really know what they're talking about... The only relevant equation I have is P=A(A^tA)^-1A^t. I don't know how to apply it. Please help!
    I'm not entirely sure what's going on either but if P=uu^{T} then P^2=(uu^{T})(uu^{T})=u(u^{T}u)u^{T}=uu^{T} since u^{T}u=u\cdot u=\|u\|^2=1
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  4. #4
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    That does help. Thanks for answering!
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