I saw this in a book as a Proposition but I think it's an error:(adsbygoogle = window.adsbygoogle || []).push({});

Assume that the (n-by-k) matrix, [tex]A[/tex], is surjective as a mapping,

[tex]A:\mathbb{R}^{k}\rightarrow \mathbb{R}^{n}[/tex].

For any [tex]y \in \mathbb{R}^{n} [/tex], consider the optimization problem

[tex]min_{x \in \mathbb{R}^{k}}\left{||x||^2\right}[/tex]

such that [tex] Ax = y[/tex].

Then, the following hold:

(i) The transpose of [tex]A[/tex], call it [tex]A^{T}[/tex] is injective.

(ii) The matrix [tex]A^{T}A[/tex] is invertible.

(iii) etc etc etc....

I have a problem with point (ii), take as an example the (2-by-3) surjective matrix

[tex]A = \begin{pmatrix}

1 & 0 & 0\\

0 & 1 & 0

\end{pmatrix}[/tex]

[tex]A^{T}A[/tex] in this case is not invertible.

Can anyone confirm that part (ii) of this Proposition is indeed incorrect ?

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# Solving for a Surjective Matrix

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