Hi Steve!
Thanks for the note. Coincidentally,
I’ve been running through quaternions lately (I’d like to understand
them – there seems to be a lot of confusion in the 3d graphics world,
where many programs expect them as input – then expand out to Clifford
algebra.)
Oh wow, I just ran through your example
(literally holding out my hand in front of me) and things started
clicking. The key intuition I’m seeing (remembering!) is that in a
matrix, each element of the new vector can be influenced by ANY of the
dimensions of the matrix.
So, the y dimensions can nudge you forward in the x direction (at some rate, or a partial rate) if it desires. The identity matrix is saying you are only impacted by the “proper” dimension at a 1:1 rate (so [x,y,z] goes in and [x, y, z] comes out).
-Kalid
On Tue, Dec 2, 2014 at 2:41 PM, Steven Lehar slehar@gmail.com wrote:
Hi Kalid,
Just a short note on an insight I had relating matrix and Clifford Algebra.
Imagine
a 3x3 matrix, now just re-arrange the terms into a 3-D structure
instead of 2. Hold out thumb, index, and middle fingers of your left
hand, as x,y,z vectors. These three values are the “main diagonal” of
the matrix. If all three are length 1, then its the identity matrix, it
changes nothing. Imagine a cloud of (x,y,z) points around your hand, the
matrix would leave them where they are.
If
you imagine your thumb, the x value, of length 2.0, this would shift
all of the points parallel to the x axis away from the origin,
“stretching” the cloud of points in the x direction only.
Now
for the off-diagonal terms, imagine a little vector from the tip of
your thumb parallel to your index finger, this represents the cross-term
how much a point is shifted in the Y direction based on its X
coordinate. If the cross term non-zero positive, then all points in the
cloud would be shifted right-and-up to the right of the origin, and
left-and-down to the left of the origin, by an amount proportional to
their x component.
Conversely,
a little vector at the tip of your index finger (Y axis) parallel to
your thumb (X axis), that represents how much points are shifted towards
the x direction based on their Y component value.
And likewise for all the other directions.
The
point is that by expressing the matrix as a 3-D spatial structure, its
operations become much more intuitive. You can visualize the matrix
operations as spatial transformations by a spatial structure, and
Clifford Algebra reveals all of algebra to be a branch of geometry.
Paul Falstad’s matrix simulation is very helpful for intuition.
http://www.falstad.com/matrix/
Steve