Why doesn't this matrix represent a rotation?

[Aziza]

The matrix is

| 1/2 -1/2 |
| 1/2 1/2 |

Why is this matrix not representing a rotation?

The form of rotation is

| cos x -sin x |
|sin x cos x |

So in this case tan x = 1 and so x = 45...isn't this rotation of 45 degrees?On similar note, for the matrix

| 5 6 |
| -6 5 |

it says it is a rotation of $\theta$ = arctan (-6/5), which I assumed was obtained by saying that cos x = 5 and sin x = -6 so tan x = -6/5...so what is the difference between this example and my problem?edit: for the second example, it actually states that this matrix is a rotation combined with scaling. Is this the main reason why the first problem can't be considered just rotation? Because it is actually also scaled? I mean that the rotation formula I gave was derived using the unit vectors, but the vectors represented by the first problem are not unit because their length is 1/2 not 1...? Idk this seems like trivial difference though..

 

[Dick]

The matrix is

| 1/2 -1/2 |
| 1/2 1/2 |

Why is this matrix not representing a rotation?

The form of rotation is

| cos x -sin x |
|sin x cos x |

So in this case tan x = 1 and so x = 45...isn't this rotation of 45 degrees?On similar note, for the matrix

| 5 6 |
| -6 5 |

it says it is a rotation of $\theta$ = arctan (-6/5), which I assumed was obtained by saying that cos x = 5 and sin x = -6 so tan x = -6/5...so what is the difference between this example and my problem?edit: for the second example, it actually states that this matrix is a rotation combined with scaling. Is this the main reason why the first problem can't be considered just rotation? Because it is actually also scaled?

Yes, sin(45 degrees)=1/sqrt(2), not 1/2. The determinant of the matrix isn't 1. What is it? It can't be a simple rotation.

 

[marty1]

sin 45 does equal cos 45, but they don't equal 1/2. That may be a rotation in an Escher drawing.

 

[Aziza]

sin 45 does equal cos 45, but they don't equal 1/2. That may be a rotation in an Escher drawing.

Then how was the angle of rotation found in the second example?

 

[Aziza]

Yes, sin(45 degrees)=1/sqrt(2), not 1/2. The determinant of the matrix isn't 1. What is it? It can't be a simple rotation.

The determinant is 1/2, but why is that significant?

 

[Dick]

Then how was the angle of rotation found in the second example?

Both matrices are a rotation times a scaling. The determinant of the matrix determines the scaling part for these examples.

 

[Aziza]

Both matrices are a rotation times a scaling. The determinant of the matrix determines the scaling part for these examples.

But then how are the angles found?

 

[Aziza]

Ohhh i see why the matrix with the 1/2 entries is not 'real' rotation...the vector is basically going from (1/2,0) to the point (1/2, 1/2), so it is not moving along a circle. But you can consider it that the vector rotated by pi/4 and scaled at the same time...is this right?

 

[Dick]

But then how are the angles found?

Factor out the square root of the determinant. If don't then you'll make statements like cos(theta)=5. There no such angle. What's left will be a rotation matrix. Then use arccos and arcsin. Then realize that you didn't have to factor the determinant out at all to get the correct arctan. BTW not all matrices can be factored into a scaling times a rotation. These are special.

 

[Dick]

Ohhh i see why the matrix with the 1/2 entries is not 'real' rotation...the vector is basically going from (1/2,0) to the point (1/2, 1/2), so it is not moving along a circle. But you can consider it that the vector rotated by pi/4 and scaled at the same time...is this right?

Right in principle, not in detail. (1,0) goes to (1/2,1/2), (1/2,0) goes to (1/4,1/4). But you've got the right idea.

 

[Aziza]

Right in principle, not in detail. (1,0) goes to (1/2,1/2), (1/2,0) goes to (1/4,1/4). But you've got the right idea.

OHHH i see! thank you so much