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Math Help - Linear Matrix question

  1. #1
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    Linear Matrix question

    if A is any mxn matrix such that m > n, is it possible to find an nxm matrix B such that BA=I_n? why or why not?

    the second question is identical just switch m<n.

    I dont understand this question at all. any help?
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  2. #2
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    Quote Originally Posted by p00ndawg View Post
    if A is any mxn matrix such that m > n, is it possible to find an nxm matrix B such that BA=I_n? why or why not?
    If A is such a matrix so that A^TA is invertible then it can be shown B= (A^TA)^{-1}A^T satisfies BA = I_n.

    For example, let A=\begin{bmatrix}1&0\\0&1\\1&1 \end{bmatrix} and follow the computations above to convince yourself.

    There is a similar formula for the second part as well.
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    Quote Originally Posted by ThePerfectHacker View Post
    If A is such a matrix so that A^TA is invertible then it can be shown B= <b>(A^TA)^{-1}</b>A^T satisfies BA = I_n.

    For example, let A=\begin{bmatrix}1&0\\0&1\\1&1 \end{bmatrix} and follow the computations above to convince yourself.

    There is a similar formula for the second part as well.

    can you explain the bolded part. What does the raised to the -1 do the transpose and A?

    woops it didnt bold, look at the [b]'s placed. sorry.
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    Quote Originally Posted by p00ndawg View Post
    can you explain the bolded part. What does the raised to the -1 do the transpose and A?

    woops it didnt bold, look at the [b]'s placed. sorry.
    oh nvm, I got it, now im just having trouble with the m<n 2nd one.

    Should I just use the same assumption but pick a different matrix in which m<n? and solve it similarly, to see what I geT?
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  5. #5
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    Quote Originally Posted by p00ndawg View Post
    if A is any mxn matrix such that m > n, is it possible to find an nxm matrix B such that BA=I_n? why or why not?

    the second question is identical just switch m<n.
    The rank of B cannot be greater than m, nor can the rank of BA. So if m<n then BA cannot be equal to I_n.
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