- #1
lauratyso11n
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I saw this in a book as a Proposition but I think it's an error:
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 ?
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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