# Matrix notation for scientific papers

I am writing a research paper for an engineering journal and I am having difficulty writing a couple of simple matrix equations.

For equation 1, I have a $$4 \times 4$$ transformation matrix $$T$$. I want to define $$\delta_x$$ where $$\delta_x$$ is the norm of the translation vector of $$T$$.

For equation 2, I have the same transformation matrix $$T$$. I want to define $$\delta_\theta$$ where $$\delta_\theta$$ is the norm of the vector $$<\alpha, \beta, \gamma >$$, and $$\alpha, \beta, \gamma$$ are the Euler angles of the $$3 \times 3$$ rotation matrix of $$T$$.

Is there a correct way to write these equations?

• You say difference but you write a product. Also the first notation $[X]_4$ seems different from the second $[X]_{a,b,c}$. Why don't you explain exactly what you want to say so that people can help you better
– lcv
May 26, 2020 at 19:12
• @Icv I edited the question to be more precise May 26, 2020 at 20:29
• Maybe you could add what you mean with "transformation matrix". May 26, 2020 at 21:55
• It's a 4x4 homogenous transformation matrix, consisting of a 3x3 rotation matrix in the upper left corner and a 3x1 translation vector in the last column. May 26, 2020 at 21:59
• May 26, 2020 at 22:19

For problem 1) I would do the following.

Let $$T$$ be the following $$4\times4$$ transformation matrix

$$T=\left(\begin{array}{cc} R & x\\ 0 & 1 \end{array}\right)$$

where $$R$$ is a $$3\times3$$ rotation (i.e. orthogonal) matrix and $$x$$ a $$3\times 1$$ (shift) vector.

Then we define

$$\delta_x = \parallel x \parallel.$$

For part 2 I would suggest:

Let now decompose $$R$$ into three elementary rotations defining Euler angles, i.e.

$$R = X(\alpha) Y(\beta) Z(\gamma) \tag{1}$$

where $$X(\phi),Y(\phi),Z(\phi)$$ represent rotations around their respective axis by an angle $$\phi$$. Let $$\theta$$ be the $$3\times 1$$ vector of angles $$\theta=(\alpha,\beta,\gamma)$$.
Then

$$\delta_\theta = \parallel \theta \parallel$$

PS

Note that I'm not sure that Eq. (1) is the definition you use for Euler angles. Another possibly more standard one is $$R= XZX$$.