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$(A^TA+aI)^{-1}A^T = A^TX$

where A and X are real-matrices and a is a positive non-zero real number. A is non-zero and you may assume that A is such that there is a unique X that satisfies the above equation.

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Using the inverse of the inverse so to speak:

$A^T=(A^TA+aI)A^TX$ $A^T=A^T(AA^T+aI)X$

and now if $X=(AA^T+aI)^{-1}$ it is solved.

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    Glad I could help!2012-10-04