Distinguishing projections, reflections and rotations from matrices

[matrix_204]

i was trying to figure out something that i didn't understand and the book doesn't have much examples of it either. My question is how do u know whether a transformation is a projection on a line, reflection on a line, or rotation through an angel? With T given. The questions i did from the book, i was able to find the line and reflection, since the same question was previously stated and i was able to do it, n got the answer from there, but didn't understand it.
Take T[x y]=1/2[x-y y-x] and by solving these i get [1 -1]/[-1 1] (this is not division, just goes at the bottom, its a 2x2 matrix, and the vectors r transposed) so by looking at this
[ 1 -1]
[-1 1]
how can u tell whether its a reflection, projection or rotation?
Similarly another result i got for another part was
[1/2 root3/2]
[root3/2 1/2 ]
i think this one is rotation but how can u tell?

 

[Galileo]

A rotation about the origin is a transformation which preserves the lengths and angles of vectors. It's called an orthogonal transformation. They are given by an orthogonal matrix, possesing the following property:
$$A^TA=I$$
where the T denotes the transpose operation.
Furthermore: if det(A)=1, then it's a rotation, if det(A)=-1, then it's a reflection followed by a rotation.
Also, a rotation over an angle $\theta$ can always be given by the following matrix:

$$\left( \begin{array}{cc}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right)$$
Since $\cos \frac{\pi}{3}=1/2$ and $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ the matrix:
$$\left( \begin{array}{cc}1/2 & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & 1/2 \end{array} \right)$$
is matrix for the rotation about an angle of $\pi/3$. (Did you forgot a minus sign in your matrix?)

Projections always have the following properties:
$$P^T=P$$
$$P^2=P$$.

 

[ehild]

Take T[x y]=1/2[x-y y-x] and by solving these i get [1 -1]/[-1 1] (this is not division, just goes at the bottom, its a 2x2 matrix, and the vectors r transposed) so by looking at this
[ 1 -1]
[-1 1]

You have omitted 1/2.

$$T=1/2 \left(\begin {array}{cc} 1&-1\\ -1& 1\end{array}\right)$$

how can u tell whether its a reflection, projection or rotation?

Find out what does this transformation do with the base vectors. You will see that they are projected onto the [1,-1] direction.

Similarly another result i got for another part was
[1/2 root3/2]
[root3/2 1/2 ]
i think this one is rotation but how can u tell?

To be a rotation, the matrix should be unitary, and yours is not (the determinant should be 1).
A matrix that represents an anti-clockwise rotation by angle alpha in the (xy) plane is

$$T=\left(\begin {array}{cc}\cos(\alpha)&-\sin(\alpha)\\ \sin(\alpha)&\cos(\alpha)\end{array}\right )$$

ehild