Problem 36 - Advanced Engineering Mathematics

Question

$$ \text { Show that if } \mathbf{A} \text { is an } m \times n \text { matrix, then } \mathbf{A A}^{T} \text { is symmetric. } $$

Step-by-Step Solution

Answer

The matrix \( \mathbf{A A}^{T} \) is symmetric because its transpose is equal to itself.

Step 1: Understanding the Problem

We need to show that the product \( \mathbf{A A}^{T} \) is symmetric. A matrix is symmetric if it equals its own transpose. So, we will prove that \( (\mathbf{A A}^{T})^{T} = \mathbf{A A}^{T} \).

Step 2: Transpose Properties

Recall that the transpose of a product of two matrices \( \mathbf{A} \) and \( \mathbf{B} \) is the product of their transposes in reversed order: \((\mathbf{AB})^T = \mathbf{B}^T \mathbf{A}^T \).

Step 3: Applying Transpose to \( \mathbf{A A}^{T} \)

Consider the transpose of \( \mathbf{A A}^{T} \). Using the property of transposes: \[(\mathbf{A A}^{T})^{T} = (\mathbf{A}^{T})^{T} \mathbf{A}^{T}\]Note that \( (\mathbf{A}^{T})^{T} = \mathbf{A} \).

Step 4: Simplification

Substitute back to simplify:\[(\mathbf{A A}^{T})^{T} = \mathbf{A} \mathbf{A}^{T}\]Thus, we have shown \((\mathbf{A A}^{T})^{T} = \mathbf{A A}^{T}\).

Step 5: Conclusion

Since \( (\mathbf{A A}^{T})^{T} = \mathbf{A A}^{T} \), \( \mathbf{A A}^{T} \) is symmetric by definition.

Key Concepts

Matrix Transpose

In linear algebra, understanding the concept of a matrix transpose is vital for comprehending various matrix operations. Transposing a matrix involves flipping it over its diagonal. This means swapping its rows and columns.
For instance, if you have a matrix \( \mathbf{A}\) with a dimension \( m \times n \), the transpose of \( \mathbf{A}\), denoted as \( \mathbf{A}^{T} \), will have dimensions \( n \times m \).
The entries of \( \mathbf{A} \) get rearranged such that the element in the \( i^{th} \) row and \( j^{th} \) column becomes the element in the \( j^{th} \) row and \( i^{th} \) column in \( \mathbf{A}^{T} \). This concept is used broadly in computations involving symmetry and orthogonality in matrices.

Properties of Transpose

The properties of transpose are crucial when working with matrices, particularly in operations like additions, multiplications, and understanding symmetrical structures. Here are some fundamental properties:

• \( (\mathbf{A}^{T})^{T} = \mathbf{A} \): The transpose of a transpose gives us back the original matrix.
• \( (\mathbf{A} + \mathbf{B})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T} \): The transpose of a sum is the sum of the transposes.
• \( (\mathbf{AB})^{T} = \mathbf{B}^{T} \mathbf{A}^{T} \): The transpose of a product of matrices is the product of their transposes in the reverse order.
• \( c\mathbf{A}^{T} = (c\mathbf{A})^{T} \): A scalar multiplied with a matrix and then transposed is the same as multiplying a scalar with the transposed matrix.

Understanding these properties helps simplify matrix operations and proofs, like showing symmetry.

Symmetric Matrix

A symmetric matrix is a fascinating concept in linear algebra, defined by the characteristic that it is equal to its transpose. This means for a matrix \( \mathbf{S} \), it holds that \( \mathbf{S} = \mathbf{S}^{T} \).
Symmetric matrices occur frequently and play a significant role in various applications, such as in systems of linear equations and matrix factorizations.
For example, if you consider \( \mathbf{A} \) as an \( m \times n \) matrix, the product \( \mathbf{A A}^{T} \) is always symmetric because it satisfies the condition:
\[ (\mathbf{A A}^{T})^{T} = \mathbf{A A}^{T} \]
This symmetry arises naturally from the properties of transposing matrix products and is independent of \( \mathbf{A} \)'s specific entries.

Transpose of a Product of Matrices

The transpose of a product of two matrices follows an intriguing rule that helps streamline many matrix calculations. For matrices \( \mathbf{A} \) and \( \mathbf{B} \), the transpose of their product is expressed as:
\[ (\mathbf{AB})^{T} = \mathbf{B}^{T} \mathbf{A}^{T} \]
This property indicates reversing the order of the matrices before taking the transpose of each.
It plays a pivotal role when proving properties like symmetry.
In the context of symmetric matrices, when proving \( \mathbf{A A}^{T} \) is symmetric, you use the transpose of a product property to arrive at:
\[ (\mathbf{A A}^{T})^{T} = (\mathbf{A}^{T})^{T} \mathbf{A}^{T} = \mathbf{A A}^{T} \]
This demonstration confirms that the transposed product equals the original product, solidifying its symmetric nature.