Ideas on what this transformation is doing?

In summary, this transformation is doing something other than simply swapping the 2nd and 3rd numbers.
  • #1
graphic7
Gold Member
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2
I'm curious to what this transformation is exactly doing. I'm lead to believe by the context of the question in my text, that this transformation is simply doing something other than "transforming". What exactly, I'm unsure.

[tex]\left(\begin{array}{ccc}0 & 0 & 1\\1 & 0 & 0\\0 & 1 & 0 \end{array}\right)[/tex]
 
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  • #2
Try it out.

Do the transformation on a vector (1,2,3), and see what happens to the result.
 
  • #3
This is an example of what is sometimes called a "fundamental matrix"- a matrix created by apply a "row operation" to the identity matrix. "Row operations" are the operations used in "Gaussian Elimination": swap two rows, multiply every number in a row by a number, add a multiple of one row to another.

Applying a "fundamental matrix" to a vector simply does the same row operation as used to create the matrix.
[tex]\left(\begin{array}{ccc}0 & 0 & 1\\1 & 0 & 0\\0 & 1 & 0 \end{array}\right)[/tex]

is created from the identity matrix by swapping the last two rows. Applying it to a vector swaps the second and third numbers.

As enigma suggests, try it on (1, 2, 3) and see what happens.
 
  • #4
Every invertible matirx is obtainable from the identity by gaussian elimination.

This particular matrix is a permutation matrix (it permutes the basis elements) which is perhaps the extra structure they are getting at.
 
  • #5
I'm just nitpicking here, but the given matrix does more than swap the 2nd and 3rd numbers. Seems like you confused it with this matrix:

[tex]\left(\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0 \end{array}\right)[/tex]
 
  • #6
the matrix rotates [tex] x,y,z [/tex] into [tex] z,x,y [/tex], or simply a rotation about the line [tex] x=y=z [/tex]. Try what enigma says. We solid state physics types recognize this as one of the generators of the [tex] O_{h} [/tex] group amonst others...
 
  • #7
Muzza said:
I'm just nitpicking here, but the given matrix does more than swap the 2nd and 3rd numbers. Seems like you confused it with this matrix:

[tex]\left(\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0 \end{array}\right)[/tex]


You're right. I need to get my eyes checked!

The given vector permutes the 3 numbers changing (1,2,3) into (3,1,2).
 
  • #8
look. a matrix is composed as follows: the first column is what happens to e1 = (1,0,0), the second column is what happens to e2 = (0,1,0), and the third column is what happens to e3 = (0,0,1).

So this matrx sends e1 to e2, e2 to e3, and ...?
 

1. What is the purpose of this transformation?

The purpose of this transformation is to change or manipulate data in a specific way. It could be used to organize, analyze, or visualize data in order to gain insights or make it more useful for a particular application or study.

2. How does this transformation affect the data?

This transformation can have various effects on the data, depending on the specific type of transformation being applied. It could alter the values or structure of the data, or it could simply reorganize the data in a different format.

3. What are the different types of transformations?

There are many different types of transformations, including mathematical transformations (such as scaling or normalization), data cleaning transformations (such as removing duplicates or filling in missing values), and data visualization transformations (such as creating charts or graphs).

4. Can this transformation be applied to any type of data?

The applicability of a transformation depends on the specific transformation and the type of data being used. Some transformations may only be relevant for certain types of data, while others can be applied to a wide range of data.

5. What are some examples of real-world applications of this transformation?

There are many potential applications of data transformations in various fields, such as finance, healthcare, marketing, and scientific research. Some examples include using transformations to identify patterns in financial data, clean and analyze medical records, segment customer data for targeted marketing, or transform and analyze data from experiments in a lab.

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