# Why do we parametrize matrices?

I'm learning about rotation matrices in class and one of the topics mentioned to parametrize a matrix. What do we need to do this?

I can't really think of a good example of when to parameterize a system of equations involving rotation matrices, but one general thing we can gain from parameterization regardless is a clear indication of the subspace that solves a linear systems of equations. For example, the video you linked shows that the solution to: $$\left[ {\begin{array}{cccccc} 1 & -4 & -2 & 0 & 3 & -5\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 1 & -4\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} } \right] \mathbf{x} = 0,\ \mathbf{x} \in \mathbb{R}^{6\times 1}$$

is the linear subspace spanned by:

$$\left[ {\begin{array}{ccc} 4 & 0 & -5 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \\ 0 & 0 & 1 \\ \end{array} } \right] \left[ {\begin{array}{c} v_1 \\ v_2 \\ v_3 \\ \end{array} } \right] \subset \mathbb{R}^{6}$$

where $$\mathbf{v} \in \mathbb{R}^{3}$$. As far as actual applications go, I suppose you could look at solutions to different robotics equations (forward/inverse kinematics) as constrained subspaces of your original space. My best guess as to why parameterization has been brought up alongside rotation matrices it to find solutions that lie in a certain subspace once a rotation has been applied. However, given the orthonormal nature of rotation matrices, you would be needing to solve for a solution subspace only given that the constraints you are trying to satisfy already lie in some linear subspace (that might be confusing, just think about how a point only has one vector "pre-rotation" that could now satisfy that point, while a line would have a line "pre-rotation" as well). If anything you would be more likely to parameterize a Jacobian or generally any other matrix that can have free variables.

To talk about a parameterizable matrix in robotics, consider a prismatic robot with two joints that both move in the z-axis only. Since you are just now learning about rotation matrices, I won't define this robot's Jacobian or anything like that - so just trust me that:

$$\mathbf{q}_{} = \left[ {\begin{array}{c} 1 \\ -1 \\ \end{array} } \right] x \subset \mathbb{R}^{2}$$

is the linear subspace of joint configurations where the robot's end-effector is in some consistent configuration - having $$\mathbf{q}$$ represents the robot's configuration and $$x \in \mathbb{R}$$. As far as revolute joint robot's go, you will only be able to use this technique at instantaneous configurations. I personally wouldn't lose much sleep over the concept, it's a cool way to think about solutions that lie in a given subspace - but techniques like singular value decomposition accomplish the same ideas in a much more relatable-to-robotics kind of way.