User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)


 * $u \ v \ \mathsf{u} \ \mathsf{v} \ \nu \ \upsilon$

Anyone else have a hard time distinguishing between $u$ and $v$? I would like it to look more like this, does it confuse anyone else? It seems PW doesn't have the upgreek package. --GFauxPas 07:49, 27 January 2012 (EST)


 * Nope. Multiple years of extensive TeX writing and reading have trained my eye. I agree that referenced $v$ looks more distinguished, but imagine it is hard to implement. --Lord_Farin 08:08, 27 January 2012 (EST)

Exponential Definitions
I am discussing the equivalence of the definitions of exponential here:

http://forums.xkcd.com/viewtopic.php?f=17&t=80256

For anyone who has been following my progress or lack thereof on exponent combination laws/log laws etc, feel free to look on. --GFauxPas 16:59, 6 February 2012 (EST)


 * Okay, it looks like $e^{xy} = e^xe^y$ was the hardest one to prove! I was expecting a walk uphill the whole way. Oh, my Linear Algebra book came in the mail, so I guess I'll work on vectors next. And one of these days I'll have to tie up loose ends with Tarski. --GFauxPas 16:57, 10 February 2012 (EST)

Matrix stuff
The purpose of this algorithm is to convert a matrix into its inverse, or to determine that such an inverse does not exist.

Let $\mathbf{A}$ be the $n \times n$ square matrix in question.

Let $\mathbf{I}$ be the identity matrix of order $n$.


 * Step 0: Create the augmented matrix $\left[{\mathbf{A} | \mathbf{I}}\right]$.


 * Step 1: Perform elementary row operations until $\left[{\mathbf{A} | \mathbf{I}}\right]$ is in reduced row echelon form. Call this new augmented matrix $\left[{\mathbf{H} | \mathbf{C}}\right]$


 * Step 2:


 * If $\mathbf{H} = \mathbf{I}$, then take $\mathbf{C} = \mathbf{A}^{-1}$.


 * If $\mathbf{H} \ne \mathbf{I}$, $\mathbf{A}$ is not invertible.

Finiteness
Follows from Matrix Row Equivalent to Reduced Echelon Matrix.

Definiteness
Follows from Matrix Row Equivalent to Reduced Echelon Matrix.

Inputs
The input is $\left[{\mathbf{A} | \mathbf{I}}\right]$ or $\mathbf{A}$, I'm not sure.

Outputs
The output is either $\left[{\mathbf{H} | \mathbf{C}}\right]$ or $\mathbf{H}$, I'm not sure.

Effectiveness
If $\mathbf{H}=\mathbf{I}$, then the effectiveness follows directly from Transformation of Identity Matrix into Inverse.

If $\mathbf{H} \ne \mathbf{I}$, then the result follows from Reduced Row Echelon Form is Unique and from Transformation of Identity Matrix into Inverse. Do I need to explain this more? Probably. On its own page?

Hence this process meets the criteria for being an algorithm, as required.