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Math Help - A transposed problem A^T

  1. #1
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    A transposed problem A^T

    let A be an m x n matrix...prove that N(A^T*A)=N(A) problem gives a hint vector x is a member of N(A^T*A) then A*vector x is a member of C(A) n N(A^T)

    B) prove that rank (A)=rank(A^T*A)

    C) prove that C(A^T*A)=C(A^T)

    Thanks
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  2. #2
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    Quote Originally Posted by bugal402 View Post
    let A be an m x n matrix...prove that N(A^T*A)=N(A) problem gives a hint vector x is a member of N(A^T*A) then A*vector x is a member of C(A) n N(A^T)

    B) prove that rank (A)=rank(A^T*A)

    C) prove that C(A^T*A)=C(A^T)

    Thanks

    What is N(A), C(A)...??

    Tonio
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  3. #3
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    n(a) is null space

    c(a) is column space

    sorry about that
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  4. #4
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    Quote Originally Posted by bugal402 View Post
    let A be an m x n matrix...prove that N(A^T*A)=N(A) problem gives a hint vector x is a member of N(A^T*A) then A*vector x is a member of C(A) n N(A^T)

    B) prove that rank (A)=rank(A^T*A)

    C) prove that C(A^T*A)=C(A^T)
    If Ax=0 then obviously A^{\textsc t}Ax=0. Therefore \mathbf{N}(A)\subseteq\mathbf{N}(A^{\textsc t}A).

    For the converse, suppose that A^{\textsc t}Ax=0. Then (Ax)^{\textsc t}(Ax) = x^{\textsc t}A^{\textsc t}Ax =0, which implies that Ax=0.

    For B), use the rank+nullity theorem.
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