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Let $A$ be the matrix:

$$\left(\begin{matrix} \alpha I_{n} \\ \beta I_{n} \end{matrix}\right)$$

where $\alpha,\beta\in\Bbb C$ are not both zero. Derive (a) the (reduced) QR factorization of $A$ and (b) the pseudoinverse of $A$.

Any help for the second question about pseudoinverse ?

Thanks in advance


I know that if 1) rank(A)=n then $A^{+} = (A^{T} A)^{-1} A^{T}$ and if 2) rank(A)=n=m then $A^{+} = A^{-1}$. I use the 1) and I found : $A^{+} = (a^{2} I_{n} + b^{2} I_{n})^{-1} $$\left(\begin{matrix} \alpha I_{n} \ \beta I_{n} \end{matrix}\right)$

note the second brackets is a matrix (1x2). How could I solve this ? Any help

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Well, you know, when $A$ has linearly independent columns, $A^+ = (A^\ast A)^{-1}A^\ast$. So ...

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