Question

Show that if \(H\) is an orthogonal matrix, then \(H^{t}(H-I)=(I-H)^{t}\). Deduce that if \(H\) is also proper, then \(\operatorname{det}(I-H)=0\). Hence show that if \(\mathcal{T}\) and \(\tilde{T}\) are two (right-handed) orthonormal triads, then there exists a nonzero vector that has the same components in both triads.

Short answer

In summary, we first showed that for an orthogonal matrix \(H\), \(H^{t}(H-I) = (I-H)\). Then, we used the fact that \(H\) is proper (having determinant equal to 1) to deduce that \(\operatorname{det}(I-H) = 0\). Finally, by considering two right-handed orthonormal triads, we demonstrated that there exists a nonzero vector that has the same components in both triads, as required.

Step-by-step solution

Show that \(H^{t}(H-I)=(I-H)^{t}\)

Given that \(H\) is an orthogonal matrix, we know that \(H^{t}H = I\). Now, we can compute the product \(H^{t}(H-I)\): $$ H^{t}(H-I) = H^{t}H - H^{t}I \implies H^{t}H - H^{t} = I - H. $$ From the properties of orthogonal matrices, we know that \(H^{t}H = I\), so this becomes: $$ I - H = (I - H). $$ This confirms that \(H^{t}(H-I) = (I-H)\) for an orthogonal matrix \(H\).

Use the fact that \(H\) is proper to deduce \(\operatorname{det}(I-H) = 0\)

Since \(H\) is a proper orthogonal matrix, its determinant is equal to 1, i.e., \(\operatorname{det}(H) = 1\). We know that \(H^{t}(H-I) = (I-H)\), which means that they have the same determinant: $$ \operatorname{det}(H^{t}(H-I)) = \operatorname{det}(I-H). $$ Using the properties of determinants, we can rewrite the left side as follows: $$ \operatorname{det}(H^{t}(H-I)) = \operatorname{det}(H^{t})\operatorname{det}(H-I). $$ Since the determinant of the transpose of a matrix is equal to the determinant of the original matrix, and the determinant of \(H\) is 1, this becomes: $$ \operatorname{det}(I-H) = 1\cdot \operatorname{det}(H-I). $$ Hence, \(\operatorname{det}(I-H) = 0\).

Show that there exists a nonzero vector with the same components in both triads

Now let's consider two right-handed orthonormal triads \(\mathcal{T}\) and \(\tilde{T}\). Since both triads are right-handed orthonormal triads, they can be represented by the proper orthogonal matrices \(H_1\) and \(H_2\), respectively. From our previous result, we know that \(\operatorname{det}(I-H_1) = 0\) and \(\operatorname{det}(I-H_2) = 0\). Therefore, there exists a nonzero vector \(v\) such that \((I-H_1)v = 0\) and \((I-H_2)v = 0\). This means that both \(H_1v\) and \(H_2v\) are equal to \(v\). This implies that the vector \(v\) has the same components in both triads \(\mathcal{T}\) and \(\tilde{T}\), concluding our proof.

Key concepts

Analytical Dynamics

Analytical dynamics is an essential branch of physics that focuses on the study of the motion of objects and the forces acting upon them. It is especially important in understanding the principles behind conservation laws, stability of motion, and oscillatory behavior in mechanical systems. In particular, analytical dynamics often employs matrices and their properties to represent and solve complex problems involving rotations and transformations.

Within this field, orthogonal matrices play a critical role. They contain orthonormal vectors as columns or rows, which represent rotations or reflection operations that preserve the length and angles of vectors. In a physical context, these matrices can describe the transformation between different reference frames or coordinate systems – a common occurrence in dynamic analysis.

Determinant of a Matrix

The determinant of a matrix is a scalar value that is a fundamental characteristic of a matrix. It provides important insights into the properties of the matrix, such as whether it is invertible or singular. In the context of orthogonal matrices, the determinant is especially significant. For an orthogonal matrix, the determinant can only take the values of +1 or -1. If the determinant is +1, the matrix is known as a proper orthogonal matrix, indicating that it represents a rotation. Conversely, if the determinant is -1, the matrix includes reflection in addition to rotation.

In our exercise, understanding the determinant is crucial when deducing that \(\operatorname{det}(I-H) = 0\) for a proper orthogonal matrix, meaning there is no change in orientation.

Orthonormal Triads

Orthonormal triads are sets of three vectors in space that are mutually perpendicular (orthogonal) and have unit length (normal). These triads form the basis of a three-dimensional coordinate system and are used to describe the orientation and position of objects in space in a clear and concise manner. An orthonormal triad is the fundamental building block of a coordinate system used in many areas of physics, including analytical dynamics.

In practical terms, if we have two orthonormal triads—that is, two different sets of orthogonal and normalized vectors—we can apply the characteristics of orthogonal matrices to understand how vectors transform between these coordinate systems. The exercise demonstrates that under these conditions, there's always a nonzero vector that will remain the same when represented in both triad systems, illustrating the consistency and reliability of orthonormal transformations in analytical dynamics.