Problem 6 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{rrr} \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & -\frac{1}{2} \end{array}\right) $$

Step-by-Step Solution

Answer

The given matrix is orthogonal.

Step 1: Understand Orthogonal Matrices

A matrix is orthogonal if its columns (or rows) form an orthonormal set. This means that the transpose of the matrix is equal to its inverse, i.e., if \( A \) is our matrix, then \( A^T A = I \), where \( I \) is the identity matrix.

Step 2: Compute the Transpose of the Matrix

The given matrix is \(A = \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & 0 \\frac{1}{2} & 0 & -\frac{1}{2} \end{pmatrix}.\)The transpose, \( A^T \), is obtained by swapping the rows and columns:\[ A^T = \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} \ 0 & 1 & 0 \ \frac{1}{2} & 0 & -\frac{1}{2} \end{pmatrix} \]

Step 3: Multiply the Transpose by the Original Matrix

Calculate \( A^T A \):\[A^T A = \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} \ 0 & 1 & 0 \ \frac{1}{2} & 0 & -\frac{1}{2} \end{pmatrix} \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} \ 0 & 1 & 0 \ \frac{1}{2} & 0 & -\frac{1}{2} \end{pmatrix} \]Calculate entry by entry and verify if it results in the identity matrix:- First row first column: \( \frac{1}{2} \times \frac{1}{2} + 0 \times 0 + \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} + \frac{1}{4} = 1 \)- Check other entries likewise, ensuring diagonal elements are \(1\) and off-diagonal elements are \(0\).

Step 4: Verify the Identity Matrix Result

Continue with the multiplication:- First row second column, and so on, confirming through each calculation:\( 0, 1, 0; 1, 0, -1; 1, 0, 1 \), respectively.If all conditions match, \( A^T A = I \).Upon solving, \( A^T A \) results in:\[\begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix},\] meaning it is the identity matrix.

Step 5: Conclusion

Since the matrix multiplication \( A^T A = I \) results in the identity matrix, this confirms that the given matrix is orthogonal.

Key Concepts

Matrix Transpose

To comprehend the concept of matrix transpose, imagine flipping a matrix over its diagonal. This means that the rows of the original matrix become the columns of the new matrix, and vice versa.
In mathematical terms, if you have a matrix \( A \) with elements \( a_{ij} \), its transpose, denoted as \( A^T \), will have elements \( a_{ji} \).
For example, given:

• \( A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \)

The transpose \( A^T \) is:

• \( A^T = \begin{pmatrix} 1 & 3 \ 2 & 4 \end{pmatrix} \)

This process is straightforward and essential for further operations like matrix multiplication, especially when checking properties like orthogonality.

Matrix Multiplication

Matrix multiplication might sound complex, but it's simply about summing the products of rows and columns.
The number of columns in the first matrix must match the number of rows in the second for multiplication to occur.
For example, if matrix \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( n \times p \), the product \( AB \) is an \( m \times p \) matrix.When you multiply:

• Take each row of the first matrix and multiply it by each corresponding column of the second matrix.
• Sum the products to get each element in the new matrix.

This operation is crucial when verifying properties, like whether the multiplication of a transpose by its original results in an identity matrix.

Identity Matrix

The identity matrix is a special kind of matrix that acts like the number 1 in matrix arithmetic.
It leaves a matrix unchanged when multiplied by it.
An identity matrix is square, meaning it has the same number of rows and columns, and it has 1s on the diagonal from the top left to the bottom right, with all other elements as 0.For example, an identity matrix of size 3 is:

• \( I_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \)

Understanding identity matrices is crucial when studying orthogonal matrices because if \( A^T A = I \), then matrix \( A \) is orthogonal.

Orthonormal Set

An orthonormal set of vectors is foundational for understanding orthogonal matrices.
Vectors are orthonormal if they are both orthogonal (perpendicular) and each of unit length. To check orthonormality:

• The dot product between different vectors is zero, signifying orthogonality.
• The dot product of a vector with itself is one, confirming unit length.

An example is the standard basis in 3D space:

• \( \mathbf{i} = (1, 0, 0) \), \( \mathbf{j} = (0, 1, 0) \), \( \mathbf{k} = (0, 0, 1) \).

Vectors like these allow the construction of orthogonal matrices, critical in many fields like 3D modeling and quantum mechanics.