Question

Consider the three-dimensional Euler rotation matrix \(\hat{R}(\phi, \theta, \psi)=\) \(\hat{R}_{z}(\psi) \hat{R}_{x}(\theta) \hat{R}_{z}(\phi)\) a. Find the elements of \(\hat{R}(\phi, \theta, \psi)\). b. Compute \(\operatorname{Tr}(\hat{R}(\phi, \theta, \psi)\). c. Show that \(\hat{R}^{-1}(\phi, \theta, \psi)=\hat{R}^{T}(\phi, \theta, \psi)\). d. Show that \(\hat{R}^{-1}(\phi, \theta, \psi)=\hat{R}(-\psi,-\theta,-\phi)\).

Short answer

The exercise explores the properties of the Euler rotation matrix in 3D. Notable results include the trace of the matrix and the proof of its orthogonal properties by validating that the transpose of the rotation matrix is its inverse. Furthermore, the exercise proves that the inverse of the rotation matrix can be obtained by negating the angles.

Step-by-step solution

Determine the Euler rotation matrix

The Euler rotation matrix in 3 dimensions is a combination of rotation matrices about z, x, and z axes respectively. By applying the Euler rotation matrices definitions about the z, x and the z axes respectively we therefore can define the root matrix \(\hat{R}\).

Compute the Trace

The trace of a matrix is the sum of its diagonal elements. By summing all diagonal elements of the \(\hat{R}\) matrix, one can compute the trace \(Tr(\hat{R})\).

Prove that \(\hat{R}^{-1} = \hat{R}^{T}\)

To prove this statement, we utilize the properties of orthogonal matrices, where the transpose of the matrix is equal to its inverse. Start by taking the transpose of \(\hat{R}\) and set this equal to \(\hat{R}^{-1}\). If both sides of the equation match, then the statement holds true.

Prove that \(\hat{R}^{-1} = \hat{R}(-\psi, -\theta, -\phi)\)

First, compute \(\hat{R}(-\psi, -\theta, -\phi)\) similarly as you did in step 1 but now with negated angles. After, equate this to \(\hat{R}^{-1}(\phi, \theta, \psi)\). If this equation holds true, the statement has been proven correctly.

Key concepts

Orthogonal Matrices

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (orthonormal vectors). Mathematically, a matrix \(Q\) is orthogonal if its transpose is equal to its inverse, which can be represented as \(Q^{T} = Q^{-1}\). This means for an orthogonal matrix \(Q\), \(Q^{T} Q = QQ^{T} = I\), where \(I\) is the identity matrix.

Orthogonal matrices are significant in many areas of mathematics and applied disciplines because they preserve the length of vectors upon multiplication and are involved in operations such as reflection, rotation, and coordinate transformations. In the case of the Euler rotation matrix, the matrix \(\hat{R}(\phi, \theta, \psi)\) is an example of an orthogonal matrix, as the inverse of the rotation is simply the transpose of the matrix, consistent with the definition of orthogonal matrices.

Matrix Trace

The trace of a matrix is a fundamental quantity in linear algebra that is defined as the sum of the elements on the main diagonal of a square matrix. In symbol, the trace of matrix \(A\) is written as \(\operatorname{Tr}(A)\). The trace is a scalar value and possesses several important properties, including the facts that it is invariant under change of basis and that the trace of a product of two matrices remains unchanged when the order of multiplication is switched if the product is square.

For a 3D Euler rotation matrix \(\hat{R}\), the trace can provide valuable information about the rotation, such as the angle of rotation, when considering the trace in relation to the characteristic polynomial of the matrix. In the given exercise, computing the trace as a summation of the diagonal elements of \(\hat{R}(\phi, \theta, \psi)\) yields a concise expression reflecting the cumulative effect of the rotations.

Matrix Transpose

The transpose of a matrix is another core concept in linear algebra and is obtained by flipping a matrix over its diagonal, which results in another matrix where the original rows become columns and vice versa. It is denoted as \(A^{T}\) if \(A\) is the original matrix. The transpose operation is crucial in the context of orthogonal matrices, such as Euler rotation matrices, because as we've previously noted, for orthogonal matrices the inverse is equal to the transpose.

This property simplifies calculations, particularly when dealing with inverse transformations in rotations and reflections. In the exercise provided, showing that \(\hat{R}^{-1}(\phi, \theta, \psi)=\hat{R}^{T}(\phi, \theta,\psi)\) involves highlighting that \(\hat{R}\) is indeed an orthogonal matrix, making use of both the properties of the transpose and the intrinsic nature of the Euler rotation matrix.

Inverse Matrices

Inverse matrices are a foundational concept in linear algebra, where the inverse of a matrix \(A\), denoted as \(A^{-1}\), is the matrix that, when multiplied with \(A\), results in the identity matrix \(I\) such that \(AA^{-1} = A^{-1}A = I\). The inverse exists only for square matrices that are nonsingular, meaning the determinant of the matrix is not zero.

In many practical applications, finding the inverse of a matrix is synonymous with solving systems of linear equations or reversing a transformation. In the context of Euler rotation matrices, the inverse represents the rotation that reverses the original rotation, which is exemplified in part d of the exercise result where \(\hat{R}^{-1}(\phi, \theta, \psi)\) is shown to be equivalent to \(\hat{R}(-\psi, -\theta, -\phi)\). This knowledge is particularly useful in areas like robotics and computer graphics, where one needs to frequently compute and apply inverse transformations.