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Given $a_1,a_2,\gamma,b_1,b_1 \in\mathbb{R}$.

Say if the Linear Least Squares problem of the following matrix has solution and if yes how many? $$ A= \begin{bmatrix} a_1 & \gamma a_1 \\ a_2 & \gamma a_1\\ \end{bmatrix} $$ $$ b= \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} $$

My answer is this: "According to one of the theorems if $A \in \mathbb{C}^{m\times n}$ and its range is maximum then its linear least squares problem has one solution and it is $A^*AX= A^*b$"

Now using the above theorem I said the problem of the least linear squares of the given matrix has unique solution since $\det(A)= \gamma a_1^2-\gamma a_1a_2 \neq 0$.

Is my solution correct? or should i have semplifed the above equation?

Thank you!

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  • $\begingroup$ What if $a_1=0$? What if $\gamma=0$? What if $a_1=a_2$? $\endgroup$ Feb 14, 2013 at 11:45
  • $\begingroup$ @GerryMyerson, Yes i should have added that condition as mentioned by Stefan Hansen below. $\endgroup$
    – tutak
    Feb 14, 2013 at 11:55

1 Answer 1

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The least squares problem has a unique solution if and only if $A$ has full rank. But since $A$ is a square matrix, then $A$ has full rank if and only if $A$ is invertible. So you're absolutely right about checking whether the determinant is non-zero. Note however, that $$ \det(A)=\gamma(a_1^2-a_1a_2) $$ is indeed $0$ in any of the following cases

  1. $\gamma=0$,
  2. $a_1=0$,
  3. $a_1=a_2$,

or any combination of these together.

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  • $\begingroup$ So i should have added that the Least Linear squares problem has a unique solution provided that $\gamma\ \neg$ 0 , $a_1 \neg$ 0 and $a_1 \neg$ a_2. right? Thank you very much for the editing and answer. $\endgroup$
    – tutak
    Feb 14, 2013 at 11:52
  • $\begingroup$ Yes (i take it that you meant $\neq$ three times). And you're welcome. $\endgroup$ Feb 14, 2013 at 11:56
  • $\begingroup$ Yes, i meant not equal :) $\endgroup$
    – tutak
    Feb 14, 2013 at 11:57

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