Refer to pg. 40

• spacecraft attitude determination is the study of methods for estimating proper orthogonal matrix that transforms vectors from a reference frame fixed in space to a frame fixed in the spacecraft body
  • dependent on the study of proper orthogonal 3 by 3 matrices (matrices in the group SO(3))
  • referred to as rotation matrices or attitude matrices
  • attitude matrices are denoted by $A$
• Euler’s Theorem: any rotation is a rotation about a fixed axis
  • transformation of a vector representation $\mathbf{x}F$ from reference frame $F$ to reference frame $G$ using an attitude matrix $A{GF}$: $A_{GF}\mathbf{x}_F=\mathbf{x}_G$
  • asserts existence of a vector $\mathbf{e}$ along the direction of the rotation axis (and has the same representation in frame $G$ as in frame $F$)
• if substituting $\mathbf{x}_F=\mathbf{x}_G=\mathbf{e}$ then Euler’s theorem becomes $A\mathbf{e}=\mathbf{e}$ (special case of eigenvalue/eigenvector relationship)

<aside> 💡 An eigenvector of a general square matrix $M$ is a nonzero vector $\mathbf{x}$ which observes the following relationship (where the eigenvalue $\lambda$ is a scalar): $M\mathbf{x}=\lambda\mathbf{x}$

</aside>

• to find the nonzero solution of $\mathbf{x}$ then the matrix $(\lambda I - M)$ must be singular otherwise the equation $\mathbf{x}=(\lambda I - M)^{-1}\mathbf{0}=\mathbf{0}$ would be valid
• using the singular matrix definition, the following is implied:

$$ \det(\lambda I - M) = \lambda^n + \alpha_1\lambda^{n-1}+ \cdots + \alpha_{n-1}\lambda + \alpha_n=0 $$

• when considering this equation of arbitrary degree $n$, this is known as the characteristic equation of $M$
  • eigenvalues are the $n$ roots of this equation (counts multiple roots as their multiplicity)
  • Euler’s theorem states that one of the eigenvalues (root of the characteristic equation) of an attitude matrix has the value $\lambda = 1$

<aside> 💡 Remember that this value of 1 corresponds to no change in the vector and obeys the equation $A\mathbf{e}=\mathbf{e}$ as explained above

</aside>

• characteristic equation of a 3 by 3 matrix $M$ is found by computation of the determinant and takes the following form:

$$ \lambda^3-\lambda^2\text{tr}M+\lambda\operatorname{tr}(\operatorname{adj}M)-\det M=0 $$

<aside> 💡 From pg. 19, note that $\operatorname{tr}A$ is the trace of the square matrix $A$ and is equal to $\operatorname{tr}A=\sum_{i=1}^n{a_{ii}}$

</aside>

<aside> 💡 From pg. 20, note that $\operatorname{adj}A$ refers to the adjoint matrix of $A$

</aside>

• for special case of proper orthogonal 3 by 3 matrix, the characteristic equation is:
  • note that there is a root of $\lambda=1$, proving the rotation axis $\mathbf{e}$

$$ 0=\lambda^3-\lambda^2\operatorname{tr}A+\lambda\operatorname{tr}A-1=(\lambda-1)^2[\lambda^2+\lambda(1-\operatorname{tr}A)+1] $$