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Math Help - pseudo inverse of a matrix

  1. #1
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    pseudo inverse of a matrix

    Hi
    Let A be an m \times n matrix. An n\times m matrix A{^\oplus} is a pseudoinverse of A if there are matrices U \ and\ V such that

    AA^{\oplus} A= A,
    A^{\oplus}=UA^{\oplus}=A^{T}V.
    Show that A^{\oplus} exists and is unique.

    Whereas i know pseudo inverse is obtained from singular value decomposition (SVD), as any m \times n matrix can be decomposed as

    A=UDV^T
    where D is diagonal matrix and U, V are orthogonal matrices.

    So we can find  A^{\oplus}=UD^{\oplus}V^T, this stratifies the first property AA^{\oplus}A = A.
    Where  D^{\oplus} is matrix of reciprocal of eigenvalues of D.
    My Question is how can

    1. How can I prove other properties A^{\oplus}=UA^{\oplus}=A^{T}V using this SVD decomposition.

    2. Here in the stated problem matrices are not orthogonal, is it OK.

    Thanks in advance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nadia321 View Post
    Hi
    Let A be an m \times n matrix. An n\times m matrix A{^\oplus} is a pseudoinverse of A if there are matrices U \ and\ V such that

    AA^{\oplus} A= A,
    A^{\oplus}=UA^{\oplus}=A^{T}V.
    Show that A^{\oplus} exists and is unique.

    Whereas i know pseudo inverse is obtained from singular value decomposition (SVD), as any m \times n matrix can be decomposed as

    A=UDV^T
    where D is diagonal matrix and U, V are orthogonal matrices.

    So we can find  A^{\oplus}=UD^{\oplus}V^T, this stratifies the first property AA^{\oplus}A = A.
    Where  D^{\oplus} is matrix of reciprocal of eigenvalues of D.
    My Question is how can

    1. How can I prove other properties A^{\oplus}=UA^{\oplus}=A^{T}V using this SVD decomposition.

    2. Here in the stated problem matrices are not orthogonal, is it OK.

    Thanks in advance
    Do you have to use the singular value decomposition (I hate it! Bad experience)? Just in case you want to google information this is often called the Moore-Penrose pseudoinverse.
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