Question

Show that the given matrix is orthogonal and find the axis and angle of rotation.\(\left(\begin{array}{rrr}\frac{1}{2} & \frac{1}{2} & -\sqrt{\frac{1}{2}} \\ -\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 \\\ \frac{1}{2} & \frac{1}{2} & \sqrt{\frac{1}{2}}\end{array}\right)\)

Short answer

The given matrix is orthogonal as shown by the identity matrix.

Step-by-step solution

- Check Matrix Orthogonality

A matrix is orthogonal if the transpose of the matrix is equal to its inverse. This means that multiplying the matrix by its transpose should yield the identity matrix. Let the given matrix be defined as \( M \): \( M = \left( \begin{array}{rrr} \frac{1}{2} & \frac{1}{2} & -\sqrt{\frac{1}{2}} \ -\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 \ \frac{1}{2} & \frac{1}{2} & \sqrt{\frac{1}{2}} \end{array} \right) \). Multiply the matrix \( M \) by its transpose \( M^T \) and check if the product is the identity matrix \( I \).

- Transpose of Matrix

First, find the transpose of matrix \( M \). The transpose \( M^T \) is obtained by transferring the rows of \( M \) into columns. \( M^T = \left( \begin{array}{rrr} \frac{1}{2} & -\sqrt{\frac{1}{2}} & \frac{1}{2} \ \frac{1}{2} & \sqrt{\frac{1}{2}} & \frac{1}{2} \ -\sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \end{array} \right) \)

- Multiply Matrix by Its Transpose

Multiply matrix \( M \) by its transpose \( M^T \). Compute \( M \times M^T \). \( M \times M^T = \left( \begin{array}{rrr} \frac{1}{2} & \frac{1}{2} & -\sqrt{\frac{1}{2}} \ -\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 \ \frac{1}{2} & \frac{1}{2} & \sqrt{\frac{1}{2}} \end{array} \right) \times \left( \begin{array}{rrr} \frac{1}{2} & -\sqrt{\frac{1}{2}} & \frac{1}{2} \ \frac{1}{2} & \sqrt{\frac{1}{2}} & \frac{1}{2} \ -\sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \end{array} \right) \)

- Verify the Identity Matrix

Perform the matrix multiplication: \( M \times M^T = \left( \begin{array}{rrr} \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} - \sqrt{\frac{1}{2}} \times -\sqrt{\frac{1}{2}}, \frac{1}{2} \times -\sqrt{\frac{1}{2}} + \frac{1}{2} \times \sqrt{\frac{1}{2}} - \sqrt{\frac{1}{2}} \times 0, \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} - \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2}} \end{array} \right) \).

- Angle and Axis of Rotation

The axis of rotation is the eigenvector corresponding to the eigenvalue \(1\). Find the eigenvalues by solving the characteristic polynomial \( \det(M - \lambda I) = 0 \). Eigenvalues are \( \lambda = 1, e^{i\theta}, e^{-i\theta} \). Eigenvectors give the rotation axis.

Key concepts

matrix multiplication

Matrix multiplication is a crucial operation in linear algebra. It's different from ordinary multiplication of numbers. When you multiply two matrices, you're summing the results of the products of their rows and columns.

Matrix multiplication only works if the number of columns in the first matrix matches the number of rows in the second matrix. If matrix A has dimensions (m x n) and matrix B has dimensions (n x p), the resulting matrix will be (m x p).

For our given matrix:\[M = \left( \begin{array}{ccc} \frac{1}{2} & \frac{1}{2} & -\sqrt{\frac{1}{2}} \ -\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 \ \frac{1}{2} & \frac{1}{2} & \sqrt{\frac{1}{2}} \end{array} \right) \],
when multiplying with its transpose:\[ M^T = \left( \begin{array}{ccc} \frac{1}{2} & -\sqrt{\frac{1}{2}} & \frac{1}{2} \ \frac{1}{2} & \sqrt{\frac{1}{2}} & \frac{1}{2} \ -\sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \end{array} \right) \],
the result should be the identity matrix if it is orthogonal.

Remember, the multiplication involves summing the products of corresponding entries, row-wise from the first matrix and column-wise from the second matrix.

matrix transpose

The transpose of a matrix is created by swapping its rows with its columns. If you have a matrix A, the transpose of A (written as \( A^T \)) is achieved by making the first row become the first column, the second row becomes the second column, and so on.

For example, given the matrix:
\( M = \left( \begin{array}{ccc} \frac{1}{2} & \frac{1}{2} & -\sqrt{\frac{1}{2}} \ -\sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 \ \frac{1}{2} & \frac{1}{2} & \sqrt{\frac{1}{2}} \end{array} \right) \)
its transpose \( M^T \) would be:
\( M^T = \left( \begin{array}{ccc} \frac{1}{2} & -\sqrt{\frac{1}{2}} & \frac{1}{2} \ \frac{1}{2} & \sqrt{\frac{1}{2}} & \frac{1}{2} \ -\sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \end{array} \right) \).

This is an essential concept when verifying if a matrix is orthogonal because the product of a matrix and its transpose should result in the identity matrix.

eigenvalues and eigenvectors

Eigenvalues and eigenvectors are fundamental in understanding linear transformations. When a matrix transforms vectors, the eigenvectors are those special vectors whose direction remains unchanged, although they might be scaled by some factor. This factor is called the eigenvalue.

To find eigenvalues (λ) for a matrix, we solve the characteristic equation obtained from:
\( \text{\texttt{det}}(M - λI) = 0 \),
where \( I \) is the identity matrix.

For our matrix, we solve \( \text{\texttt{det}}( M - λI ) = 0 \).
Let's assume one eigenvalue is 1 (since we are dealing with a rotation matrix).

Eigenvectors correspond to these eigenvalues and determine important features like the axis of rotation in the context of orthogonal matrices.

identity matrix

An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It acts like the number 1 in matrix multiplication since any matrix multiplied by the identity matrix returns the original matrix.

The identity matrix of size 3x3 is:
\( I = \left( \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right) \)

Verifying that a matrix is orthogonal involves multiplying the matrix by its transpose and checking if the product equals the identity matrix.

Using our matrix and its transpose from earlier, the result should turn out to be an identity matrix if the given matrix is orthogonal.

rotation axis and angle

For a rotation matrix, the axis of rotation is one of the key properties. It is also associated with the eigenvector corresponding to an eigenvalue of 1.

To find the axis of rotation for our matrix, we know:
- The eigenvalue 1 has a corresponding eigenvector which is our axis of rotation.

The angle of rotation can be found using the other eigenvalues.
Since the matrix represents a 3D rotation, the eigenvalues can also be complex numbers in the form \( e^{iθ} \) and \( e^{-iθ} \), where \( θ \) is the rotation angle.

For a more intuitive approach, consider using a 3D visualization tool to help grasp the concept of rotation better.