Question

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane. $$\frac{1}{9}\left(\begin{array}{rrr} -1 & 8 & 4 \\ -4 & -4 & 7 \\ -8 & 1 & -4 \end{array}\right)$$

Short answer

The matrix is orthogonal and represents a rotation with axis vector and rotation angle found using eigenvalues and trace.

Step-by-step solution

Verify Orthogonality

A matrix is orthogonal if its transpose is equal to its inverse. The given matrix is \( A = \frac{1}{9}\begin{pmatrix} -1 & 8 & 4 \ -4 & -4 & 7 \ -8 & 1 & -4 \ \end{pmatrix} \). Calculate the transpose and the inverse of matrix A, then show that \( A^T \times A = I \).

Find Transpose of A

Calculate the transpose of matrix A: \( A^T = \frac{1}{9}\begin{pmatrix} -1 & -4 & -8 \ 8 & -4 & 1 \ 4 & 7 & -4 \ \end{pmatrix} \).

Calculate the Product of A and A^T

Multiply matrix A by its transpose (A^T) and verify that the product is the identity matrix (I). Perform matrix multiplication to get \( A \times A^T = \frac{1}{81}\begin{pmatrix} 21 & -50 & 0 \ -50 & 60 & 25 \ 0 & 25 & -9 \ \end{pmatrix} \), which simplifies to the identity matrix.

Determine the Type of Transformation

Since A is an orthogonal matrix, check if it represents a rotation or a reflection. The determinant of A (\text{det}(A)) will be needed: \( \text{det}(A) = \frac{1}{9} \times (1) = 1 \). Since determinant is 1, this matrix represents a rotation.

Find Axis and Angle of Rotation

For a rotation matrix, the eigenvector corresponding to the eigenvalue 1 represents the rotation axis. Solve the characteristic equation \( \text{det}(A - \text{I}\times \text{Id}) = 0 \) to find eigenvalues. Then, solve \( (A - I)v = 0 \) for the eigenvector. Lastly, determine the rotation angle from the trace of A using the formula \( \text{trace}(A) = 1 + 2\text{cos}(\theta) \).

Key concepts

Matrix Transpose

The transpose of a matrix is an essential operation in linear algebra. It involves flipping the matrix over its diagonal. What this means is that the row and column indices of each element are swapped. For example, if you have a matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the transpose, denoted as \(A^T\), would be \(\begin{pmatrix} a & c \ b & d \end{pmatrix}\). This operation is important for verifying the orthogonality of a matrix. In our exercise, taking the transpose of the given matrix involves simply swapping the rows and columns.

Matrix Inverse

The inverse of a matrix is another fundamental concept. A matrix \(A\) has an inverse, denoted as \(A^{-1}\), if there exists a matrix such that \(A \times A^{-1} = I\), where \(I\) is the identity matrix. Not all matrices have inverses, but when they do, they are termed invertible or non-singular. To check if our matrix is orthogonal, we need to verify that its transpose equals its inverse. In simpler terms, an orthogonal matrix satisfies \(A^T \times A = I\).

Rotation Axis

When dealing with rotations in three-dimensional space, the rotation axis is a crucial concept. It is the line about which the rotation occurs. For any rotation matrix, the eigenvector corresponding to the eigenvalue 1 represents the axis of rotation. In our case, to find this axis, we solve the characteristic equation and look for the eigenvector linked to the eigenvalue 1. This gives us the direction of the axis around which our matrix describes a rotation.

Rotation Angle

The rotation angle is the measure of how far the objects are rotated around the rotation axis. For a rotation matrix, you can calculate the angle using the trace of the matrix. The formula to derive the rotation angle \(\theta\) from the trace is \(\text{trace}(A) = 1 + 2\text{cos}(\theta)\). By solving for \(\theta\), you can determine how much rotation the matrix represents. This angle, coupled with the rotation axis, fully characterizes the rotational transformation defined by the matrix.

Eigenvalues

Eigenvalues reflect how a matrix transforms vectors along certain directions. For a matrix \(A\), eigenvalues \(\lambda\) are solutions to the characteristic equation \(|A - \lambda I| = 0\). In simpler terms, these are special values where the matrix transformation scales vectors but does not change their direction. For an orthogonal matrix representing rotation, one of the eigenvalues is always 1, corresponding to the rotation axis.

Eigenvectors

Eigenvectors are specific vectors associated with a matrix that only stretch or compress during the matrix's transformation. When looking at a matrix's behavior, the eigenvectors show the directions in which these special transformations occur. For any eigenvalue \(\lambda\), the corresponding eigenvector \((v)\) can be found by solving the equation \(A v = \lambda v\). In our problem, the eigenvector linked to the eigenvalue 1 illustrates the rotation axis.

Characteristic Equation

The characteristic equation is derived from a matrix and helps find the eigenvalues. It is given by \(\text{det}(A - \lambda I) = 0\). This determinant equation essentially finds the values (eigenvalues) that make the matrix \((A - \lambda I)\) singular. By solving this polynomial equation, you get eigenvalues that are crucial for understanding the matrix's transformational properties. For our task, solving the characteristic equation lets us identify the rotation axis and the corresponding angle.