Question

Construct an orthogonal matrix using the skew-symmetric matrix $$ \mathrm{A}=\left|\begin{array}{ccc} 0 & -2 & 1 \\ 2 & 0 & -5 \\ -1 & 5 & 0 \end{array}\right| $$ (a) \(\quad(1 / 31)\left|\begin{array}{ccc}21 & 6 & 22 \\ 14 & 21 & 6 \\ 18 & 14 & 21\end{array}\right|\) (b) \(\quad(1 / 31)\left|\begin{array}{ccc}21 & 6 & 22 \\ 14 & -27 & -6 \\ 18 & 14 & -21\end{array}\right|\) (c) \(\quad(1 / 31)\left|\begin{array}{ccc}21 & 6 & 22 \\ 6 & 14 & 18 \\ 22 & 18 & 7\end{array}\right|\) (d) \(\quad(1 / 31)\left|\begin{array}{ccc}21 & -6 & 22 \\ 6 & 14 & -18 \\\ -22 & 18 & 7\end{array}\right|\)

Short answer

Using an approximation method, we have calculated the matrix exponential of the skew-symmetric matrix A, which resulted in an orthogonal matrix that doesn't match any of the given options. Eq. (4) can be expressed as: $$ e^A \approx \frac{1}{62} \left( \begin{array}{ccc} 217 & -124 & 0 \\ 124 & 1868 & -310 \\ 0 & 310 & 1674 \end{array} \right) $$ This answer is based on the approximation method used, and a more accurate method or additional terms in the expansion may lead to a matching option.

Step-by-step solution

Compute the square of the skew-symmetric matrix A

First, we compute the square of the skew-symmetric matrix A by matrix multiplication: $$ A^2= \left( \begin{array}{ccc} 0 & -2 & 1 \\ 2 & 0 & -5 \\ -1 & 5 & 0 \end{array} \right) \left( \begin{array}{ccc} 0 & -2 & 1 \\ 2 & 0 & -5 \\ -1 & 5 & 0 \end{array} \right) = \left( \begin{array}{ccc} 5 & 0 & -11 \\ 0 & 29 & 0 \\ 11 & 0 & 26 \end{array} \right) $$

Calculate the matrix exponential of A

To compute the matrix exponential of A (denoted as \(e^A\)), we calculate an infinite series given by the formula \(e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots \). But since A is a skew-symmetric matrix, successive higher powers of A will have rapidly decreasing magnitudes, and after a certain power, they can be considered as close to zero. So, we consider only the first few terms up to the square (A^2) in the expansion, giving us an approximation: $$ e^A \approx I + A + \frac{A^2}{2!} $$

Substitute the matrix values

Substitute the matrices A, and A^2 in the approximation formula: $$ e^A \approx \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) + \left( \begin{array}{ccc} 0 & -2 & 1 \\ 2 & 0 & -5 \\ -1 & 5 & 0 \end{array} \right) + \frac{1}{2} \left( \begin{array}{ccc} 5 & 0 & -11 \\ 0 & 29 & 0 \\ 11 & 0 & 26 \end{array} \right) = \frac{1}{62} \left( \begin{array}{ccc} 62 & 0 & 0 \\ 0 & 62 & 0 \\ 0 & 0 & 62 \end{array} \right)+ \frac{1}{62} \left( \begin{array}{ccc} 0 & -124 & 62 \\ 124 & 0 & -310 \\ -62 & 310 & 0 \end{array} \right) + \frac{1}{124} \left( \begin{array}{ccc} 155 & 0 & -341 \\ 0 & 1806 & 0 \\ 341 & 0 & 1612 \end{array} \right) $$ Now, add the three matrices: $$ e^A \approx \frac{1}{62} \left( \begin{array}{ccc} 217 & -124 & 0 \\ 124 & 1868 & -310 \\ 0 & 310 & 1674 \end{array} \right) $$

Compare the computed matrix with the options

Comparing the computed matrix with the options (a), (b), (c), and (d), we see that none of them match our computed matrix. However, it's important to note that we used an approximation to calculate the matrix exponential. It is possible that with a more accurate method or considering additional terms in the expansion, the result may match one of the given options. For now, with the approximation we have used, none of the given options match the computed orthogonal matrix.

Key concepts

Skew-Symmetric Matrix

Skew-symmetric matrices are squared matrices that are equal to the negative of their transpose. In precise mathematical language, a matrix A is skew-symmetric if
\( A^T = -A \). This property implies that all diagonal entries of a skew-symmetric matrix must be zeros since they would be equal to their own negatives. In the given problem, the matrix
\( \mathrm{A} = \left|\begin{array}{ccc} 0 & -2 & 1 \ 2 & 0 & -5 \ -1 & 5 & 0 \end{array}\right| \)
is skew-symmetric. Note that such matrices often arise in areas like angular velocity and certain types of linear transformations that preserve volume.

Understanding the skew-symmetry of a matrix is essential before delving into operations like matrix square and exponentials, as skew-symmetry imposes particular patterns upon the results of these operations.

Matrix Exponential

The matrix exponential is a function on matrices that is somewhat analogous to the exponential function for real numbers. For a matrix A, the exponential is denoted as
\( e^A \), and it is defined using the power series:
\( e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots \), where I is the identity matrix. Interestingly, when matrix A is skew-symmetric, the matrix exponential
\( e^A \) often results in an orthogonal matrix, which is a matrix whose transpose is its own inverse.

Calculating the matrix exponential directly using the power series can become complex, hence approximations are common in practical applications. As seen in the step-by-step solution, we limit the series to
\( e^A \approx I + A + \frac{A^2}{2!} \) to find an approximate exponential. These approximations are particularly useful when analyzing systems in physics and engineering that require an orthogonal state represented by the matrix exponential of a skew-symmetric matrix.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce another matrix. This operation is not commutative, meaning that AB is not necessarily equal to BA. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix has dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

During the multiplication, each entry at position (i, j) of the resulting matrix is calculated by taking the dot product of the ith row of the first matrix and the jth column of the second matrix. In the exercise, we see matrix multiplication in action when computing the square of the skew-symmetric matrix A, which is an essential step required for approximating the matrix exponential. This step intricately weaves together both the concepts of skew-symmetry and matrix multiplication to further our understanding of matrix exponentials and their properties.