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Thread: Projector and inverse of a vector.

  1. #1
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    Projector and inverse of a vector.

    Consider the matrix A=uv* where u and v lie in C^n (C:complex numbers). Under what condition on u and v is A a projector?

    I know that a projector is a square matrix P for which P^2= P.
    Now according to this definition, would that make uv*=Identity. and then u be the inverse of v*?

    Also what is the inverse of a vector that belongs to C^n? I mean what is u^-1? Is u^-1 a vector such that uu^-1 = the (nxn) identity matrix ?
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  2. #2
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    Quote Originally Posted by math8 View Post
    Consider the matrix A=uv* where u and v lie in C^n (C:complex numbers). Under what condition on u and v is A a projector?

    I know that a projector is a square matrix P for which P^2= P.
    Now according to this definition, would that make uv*=Identity. and then u be the inverse of v*?

    Also what is the inverse of a vector that belongs to C^n? I mean what is u^-1? Is u^-1 a vector such that uu^-1 = the (nxn) identity matrix ?
    "inverse" is only defined for functions and operators, not for vectors. It makes no sense to say that a vector is "inverse" to another.

    In any case, if P= uv* and P^2= P, it does not follow that uv*= Identity. Projectors, in general, are not the identity operator.
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  3. #3
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    Ok it makes sense, but then what are the conditions on u and v for uv* to be a projector?
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