Identity Matrix
Inverse Matrix
Product of a Matrix and a Vector
Rotations
Rotation-Dilations
Orthogonal Projections
Reflections
Determinant and Inverse of a 2x2 Matrix
Matrix ProductsHow to Examples Exercise
How to:
Rotation-Dilation matrices are similar to the Rotation matrices we saw in the last section except for one key difference: these matrices also stretch or shrink the length of the vector by which it is multiplied.
These matrices commonly take the form:
A = [b -c; c b]
Where, in polar coordinates:
b = r*cos(a),
c = r*sin(a),
r = sqrt(b^2 + c^2) and
tan(a) = c/b
This cause a counter-clockwise rotation through the angle 'a' and follow by a dilation of r. Depending on the value of 'r', our vector will:
1) Stretch by a factor of 'r', if r > 1
2) Remain the same, if r == 1
3) Shrink by a factor of 'r', if 0 < r < 1
4) Disappear, if r == 0
5) Shrink and reflect across both axes (180 degrees), if -1 < r < 0
6) Reflect across both axes but maintain length, if r == -1
or 7) Stretch and reflect across both axes, if r < -1
Examples:
Let's consider the rotation-dilation matrix, A = [1 -1; 1 1] and the vector, x = [1; 1].
'x' is a vector that makes a 45 degree angle with the positive-x axis and has length sqrt(2).
'A' is a matrix where tan(a) = 1/1, so our rotation is one of 45 degrees, and r = sqrt(1^2 + 1^2), so our dilation is sqrt(2).
So the resulting vector is rotated 45 degrees, thus lining it up with the positive-y axis, and its length is stretched by a factor of sqrt(2), giving it a total length of 2. The answer is y = [0; 2]. Letting MATLAB do the computation for us:
y = A*x
This gives us the same answer, but now we understand where it came from.
Exercise: