Recent content by lauratyso11n

And THIS is your explanation to the question: "How does light know which path to take to travel across non-homogenous media in the shortest possible time ?"

The correct statement is that if A is surjective then A^T is injective and AA^T is invertible. The formula for the optimal x is \hat{x}=A^T(AA^T)^{-1}y

SORRYYYYYY, the A^* is actually an A^T. I've corrected it.

I don't think it's a typo as he uses this result to carry some other analysis further. The crux of it is: \sigma\lambda = \alpha - \bf{r} where \sigma \in \mathbb{R}^{n-by-k} is surjective, and \lambda \in \mathbb{R}^{k}, \alpha , \bf{r} \in \mathbb{R}^{n}. How would you solve for \lambda ...

The full Proposition is as follows: Assume that the (n-by-k) matrix, A, is surjective as a mapping, A:\mathbb{R}^{k}\rightarrow \mathbb{R}^{n}. For any y \in \mathbb{R}^{n} , consider the optimization problem min_{x \in \mathbb{R}^{k}}\left{\left||x|\right|^2\right} such that Ax = y...

That would be fine, but the result is used to carry some analysis. The crux of it is: \sigma\lambda = \alpha - \bf{r} where \sigma \in R^{n-by-k} is surjective, and \lambda \in R^{k}, \alpha , r \in R^{n}. How would you solve for \lambda ? Isn't is critical that the 'typo' has to be...

I saw this in a book as a Proposition but I think it's an error: Assume that the (n-by-k) matrix, A, is surjective as a mapping, A:\mathbb{R}^{k}\rightarrow \mathbb{R}^{n}. For any y \in \mathbb{R}^{n} , consider the optimization problem min_{x \in \mathbb{R}^{k}}\left{||x||^2\right} such...

I saw this in a book as a proposition but I think it's an error: Assume that the (n-by-k) matrix, A, is surjective as a mapping, A:R^{k}\rightarrow R^{n}. For any y \in R^{n} , consider the optimization problem min_{x \in R^{k}}\left{||x||^2\right} such that Ax = y. Then, the following...