Testwiki:Reference desk/Archives/Mathematics/2011 September 17

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September 17

Normalizing logarithmic lines

I have a bunch of logarithmic equations in the form y=a ln(x)+b. In some cases, a and b are in the range of 1,000,000 to 10,000,000. In others, they are on the range of 100 to 1,000. Is there a "proper" way to normalize these so that they can be graphed together and compared. I am looking at similarities in the curve. Because of my requirement, I considered graphing y=(a/b) ln(x), which would make the y intercept the same for all of the graphs. Note: The x-scale is the same for all graphs: 1 to 100. -- kainaw™ 19:23, 17 September 2011 (UTC)

Update: Because I'm looking at probability distributions with vastly different population sizes, I think that using Zipf–Mandelbrot law may be a better method of comparison than comparing the logarithmic trend lines. The trick is to convert something like f(x)=-10278.8 ln(x)+48873.6 to a Zipfian form. -- kainaw™ 19:34, 17 September 2011 (UTC)

Finding the transformations of a matrix

Moved from Wikipedia talk:WikiProject Mathematics

The paragraph http://en.wikipedia.org/wiki/Transformation_matrix#Finding_the_matrix_of_a_transformation explains how to find the matrix belonging to a linear map. But how do I find the transformations that belong to a given matrix, which means finding the angle of rotation, scale factor and so on for the basic transformations? In other words: how to decompose a matrix into the basic transformations mentioned in said paragraph. — Preceding unsigned comment added by 84.157.37.3 (talk) 16:01, 17 September 2011 (UTC)

You might try Jordan normal form, especially the section on real matrices.--RDBury (talk) 20:02, 17 September 2011 (UTC)

I'm not sure that that's what the OP's after. Putting a matrix into Jordan normal form is done by changing the basis. I think the OP wants to write a given linear transformation, with respect to a given basis, as the composition of a rotation, a shear transformation, a reflection, a dilation, etc. I'm not sure that there's a unique answer either. Matrix multiplication isn't commutative and any fixed matrix can be written as uncountably many products of pairs of matrices. — Fly by Night (talk) 21:15, 17 September 2011 (UTC)

If the 2x2 matrix in question is of the form $e^X$, then

$X=a\left[\begin{array}{ccc}1& & 0\\ 0& & 1\end{array}\right]+b\left[\begin{array}{ccc}1& & 0\\ 0& & -1\end{array}\right]+c\left[\begin{array}{ccc}0& & 1\\ 1& & 0\end{array}\right]+d\left[\begin{array}{ccc}0& & -1\\ 1& & 0\end{array}\right]$

Here $e^a$ is the dilation factor, $e^b$ and $e^c$ are shear factors, and d is the rotation angle. Reflections are not of the form $e^X$, I think. Bo Jacoby (talk) 23:58, 17 September 2011 (UTC).

Any matrix can be (basically uniquely) written as a strain followed by a (improper) rotation. This is the polar decomposition. Sławomir Biały (talk) 00:18, 18 September 2011 (UTC)