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Given a matrix A (possibly singular), a matrix B and a vector $\mathbf{y}$, I need to compute:

  1. the last row of the matrix $$ A^+B $$

  2. the last entry of the vector $$ A^+\mathbf{y} $$

Is my only option that of calculating the pseudo-inverse $A^+$, compute the inner product and then take the last row/entry? Or is there a shortcut?

Ziofil
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  • Both questions boil down to the question how to calculate the last row of the pseudoinverse – Andreas H. Sep 27 '22 at 15:46
  • why the downvolte? what is wrong with this question? – Andreas H. Sep 27 '22 at 16:03
  • You want to calculate $e_n^TA^+B$. You may calculate $x^T=e_n^TA^+$ first and then calculate the vector-matrix product $x^TB$. So, you need to find not just any, but the minimum-norm least-square solution $x$ to $A^Tx=e_n$. I think the conjugate gradient method is able to do this. Apparently some variants of gradient descent is also capable of finding the minimum-norm least square solution. – user1551 Sep 27 '22 at 17:09

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