To calculate a rotation matrix between two coordinate systems, defined by a start matrix A and a desired matrix B, the steps below outline the process:

Steps for Calculating the Rotation Matrix:

1. Define the Matrices: Let $A$ and $B$ represent the initial and desired coordinate systems, respectively. Each matrix should be a 3x3 matrix where each column vector represents the basis vectors of the respective coordinate system.

   \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}, \quad B = \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{bmatrix}$$ where $$\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$$ are the basis vectors of the start matrix $$A$$, and $$\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3$$ are the basis vectors of the desired matrix $$B$$.

2. Rotation Matrix Concept: The rotation matrix $R$ that transforms the coordinate system from $A$ to $B$ is given by: $R = B A^{-1}$ where $A^{-1}$ is the inverse of the matrix $A$.

   The idea is that multiplying by $A^{-1}$ undoes the rotation in the original system, and multiplying by $B$ applies the new desired rotation.

3. Calculate the Inverse of $A$: To find $A^{-1}$, the inverse of the start matrix, use standard matrix inversion techniques:

   • If $A$ is orthogonal (i.e., its column vectors are orthonormal), $A^{-1} = A^\top$, the transpose of $A$.
   • Otherwise, compute the inverse of $A$ using Gaussian elimination, adjugate/cofactor method, or direct matrix inversion formulas.

4. Multiply the Matrices: Once you have $A^{-1}$, multiply it by the matrix $B$: $R = B A^{-1}$ Perform standard matrix multiplication to get the rotation matrix $R$.

5. Interpret the Result: The resulting matrix $R$ is the rotation matrix that transforms the coordinate system from $A$ to $B$. This matrix can be used to rotate vectors from the start system into the desired system.

Example:

If $A$ and $B$ are defined as follows:

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Here, $$A$$ is the identity matrix (standard basis), and $$B$$ represents a 90° counterclockwise rotation about the z-axis. 1. The inverse of $$A$$ is: \[ A^{-1} = A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix