Problem 6 - Advanced Engineering Mathematics

Question

Determine whether the given matrix is orthogonal. $$ \left(\begin{array}{rrr} \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & -\frac{1}{2} \end{array}\right) $$

Step-by-Step Solution

Answer

The given matrix is not orthogonal.

Step 1: Define Orthogonal Matrix

A matrix is orthogonal if the dot product of its rows (or columns) results in an identity matrix. This means that the transpose of the matrix is equal to its inverse.

Step 2: Calculate Transpose of Matrix

Compute the transpose of the given matrix. For matrix \[A = \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & 0 \\frac{1}{2} & 0 & -\frac{1}{2}\end{pmatrix}\]the transpose is \[A^T = \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & 0 \\frac{1}{2} & 0 & -\frac{1}{2}\end{pmatrix}\] (which happens to be the same in this case because the first and third rows are reflections of each other across the diagonal).

Step 3: Multiply Matrix by Its Transpose

Multiply the original matrix by its transpose to check if it results in an identity matrix. \[A \times A^T = \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & 0 \\frac{1}{2} & 0 & -\frac{1}{2}\end{pmatrix}\times \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & 0 \\frac{1}{2} & 0 & -\frac{1}{2}\end{pmatrix}\]Calculating each element of the resulting matrix, we determine the product.

Step 4: Verify the Result of Multiplication

Calculate each element of the product to check if it matches an identity matrix. The calculations for the elements of \(A\times A^T\) are:- First row, first column: \( \left(\frac{1}{2}\right)^2 + 0 + \left(\frac{1}{2}\right)^2 = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \)- First row, second column: \( 0 \times \frac{1}{2} + 1 \times 0 + 0 \times -\frac{1}{2} = 0 \)- First row, third column: \( \left(\frac{1}{2}\right) \times \frac{1}{2} + 0 + \left(\frac{1}{2}\right) \times -\frac{1}{2} = 0 \)Continuing with these calculations confirms that the resulting product is not an identity matrix.

Step 5: Conclude Orthogonality

Since the product of the matrix with its transpose did not result in an identity matrix, the given matrix is not orthogonal.

Key Concepts

Matrix Transpose

Transposing a matrix is like flipping it over its diagonal. This means swapping its rows and columns. If you have a matrix \(A\) with elements labeled as \(a_{ij}\), then its transpose, expressed as \(A^T\), will take the element from row \(i\), column \(j\) of \(A\) and place it in row \(j\), column \(i\) of \(A^T\). Simple, right?

• To transpose a 3x3 matrix, exchange the elements as follows: the first row becomes the first column, the second row becomes the second column, and so on.
• For symmetric matrices, the transpose looks the same as the original. This happens when the matrix is mirrored over its diagonal.

In our context, transposition helps verify the orthogonality of matrices. An orthogonal matrix remains unchanged when multiplied by its transpose, eventually resulting in an identity matrix. So, understanding transposition is crucial for identifying orthogonal matrices.

Matrix Multiplication

Matrix multiplication might seem complex at first, but it follows a systematic process. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. Then, each element in the resulting matrix is the sum of the products of the corresponding entries.

• Consider two matrices, \(A\) (of size \(m \times n\)) and \(B\) (of size \(n \times p\)). The product, \(C = A \times B\), will have dimensions \(m \times p\).
• To calculate an entry \(c_{ij}\) in \(C\), you multiply each element of the \(i\)-th row of \(A\) by the corresponding element in the \(j\)-th column of \(B\) and find their sum.

This multiplication is essential for checking whether a matrix is orthogonal. By multiplying a matrix by its transpose, we can check if the product is the identity matrix. If it is, the original matrix is orthogonal.

Identity Matrix

Imagine the identity matrix as the matrix version of the number 1 in arithmetic. It's a special kind of square matrix where the diagonal elements are all 1s, and off-diagonal elements are 0s. This unique setup causes any matrix \(A\), when multiplied by the identity matrix \(I\), to remain unchanged: \(A \times I = A\).

• For a 3x3 identity matrix: \[ I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]
• Because of its properties, the identity matrix plays a crucial role in defining orthogonality. If a matrix \(A\) is orthogonal, the product \(A \times A^T\) will yield the identity matrix.

Understanding the identity matrix helps in dealing with tasks like verifying orthogonal matrices. In essence, it serves as the benchmark for confirming whether the result of a particular matrix operation is correct or not.