Ideas on what this transformation is doing?

[graphic7]

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.

$$\left(\begin{array}{ccc}0 & 0 & 1\\1 & 0 & 0\\0 & 1 & 0 \end{array}\right)$$

 

[enigma]

Try it out.

Do the transformation on a vector (1,2,3), and see what happens to the result.

 

[HallsofIvy]

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.
$$\left(\begin{array}{ccc}0 & 0 & 1\\1 & 0 & 0\\0 & 1 & 0 \end{array}\right)$$

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.

 

[matt grime]

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.

 

[Muzza]

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:

$$\left(\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0 \end{array}\right)$$

 

[Dr Transport]

the matrix rotates $$x,y,z$$ into $$z,x,y$$, or simply a rotation about the line $$x=y=z$$. Try what enigma says. We solid state physics types recognize this as one of the generators of the $$O_{h}$$ group amonst others...

 

[HallsofIvy]

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:

$$\left(\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0 \end{array}\right)$$

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).

 

[mathwonk]

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 ...?