Symmetric matrix transform on space
Symmetric matrices are a very important family of matrices that appear very often in machine learning literature.
We know what influence diagonal and orthogonal matrices have on an input space.
A symmetric matrix could sometimes be diagonal or orthogonal, so we already know what to expect in those situations.
In the next demo, we have constrained the first element of the second row vector to move in sync with the second element of first row vector.
This ensures a symmetric matrix at all times.
Keep a watch on the dark blue dots lining the \(X\)-axis as you move the first row of the matrix.
These points rotate along with that vector.
Along notice the light green dots. These always align with the second row vector.
Also note that when the two row vectors are aligned along the same plane, the space is squashed into a single dimension.
So all points fall into a line and the line is along the same direction as the direction of the first row.
Every time the two row vectors align the space flips over, as the vectors cross each other.
Now stabilize the first row and see what happens when you move the second row along its constrained plane.
You will notice that there is no more rotation.
Instead, there is stretching, shrinking, and inverting across the plane set up by the first row.
It is almost like the first row vector is setting the direction (rotation) of the space and then the second dependent vector is acting as a diagonal matrix along that orientation.
Try out the demo to see if you can visualize these ideas.