Problem 21 - Advanced Engineering Mathematics

Question

An \(n \times n\) matrix \(\mathbf{B}\) is symmetric if \(\mathbf{B}^{T}=\mathbf{B}\), and an \(n \times n\) matrix \(\mathbf{C}\) is skew- symmetric if \(\mathbf{C}^{T}=-\mathbf{C}\). By noting the identity \(2 \mathbf{A}=\mathbf{A}+\mathbf{A}^{T}+\mathbf{A}-\mathbf{A}^{T}\), show that any \(n \times n\) matrix \(\mathbf{A}\) can be written as the sum of a symmetric matrix and a skewsymmetric matrix.

Step-by-Step Solution

Answer

Any matrix \( \mathbf{A} \) can be written as \( \mathbf{A} = \frac{1}{2}(\mathbf{A} + \mathbf{A}^{T}) + \frac{1}{2}(\mathbf{A} - \mathbf{A}^{T}) \).

Step 1: Understanding Matrix Definitions

A symmetric matrix satisfies the property: \( \mathbf{B}^{T} = \mathbf{B} \). A skew-symmetric matrix satisfies \( \mathbf{C}^{T} = -\mathbf{C} \). This means if we transpose a symmetric matrix, we get the same matrix back, while transposing a skew-symmetric matrix gives us the negative of the original matrix.

Step 2: Decompose the Matrix Expression

The given identity is: \( 2 \mathbf{A} = \mathbf{A} + \mathbf{A}^{T} + \mathbf{A} - \mathbf{A}^{T} \). Rearranging this, it can be seen that \( \mathbf{A} = \frac{1}{2} (\mathbf{A} + \mathbf{A}^{T}) + \frac{1}{2} (\mathbf{A} - \mathbf{A}^{T}) \). This looks like a decomposition of \( \mathbf{A} \) into two components.

Step 3: Check Symmetric Component

The first component, \( \frac{1}{2} (\mathbf{A} + \mathbf{A}^{T}) \), is symmetric. This is because \( (\mathbf{A} + \mathbf{A}^{T})^{T} = \mathbf{A}^{T} + \mathbf{A} = \mathbf{A} + \mathbf{A}^{T} \), showing it equals its transpose.

Step 4: Check Skew-Symmetric Component

The second component, \( \frac{1}{2} (\mathbf{A} - \mathbf{A}^{T}) \), is skew-symmetric. This follows as \((\mathbf{A} - \mathbf{A}^{T})^{T} = \mathbf{A}^{T} - \mathbf{A} = -(\mathbf{A} - \mathbf{A}^{T}) \), confirming it is the negative of its transpose.

Step 5: Combine the Components

The decomposition \( \mathbf{A} = \mathbf{S} + \mathbf{C} \) where \( \mathbf{S} = \frac{1}{2} (\mathbf{A} + \mathbf{A}^{T}) \) and \( \mathbf{C} = \frac{1}{2} (\mathbf{A} - \mathbf{A}^{T}) \) shows that any matrix \( \mathbf{A} \) can indeed be expressed as the sum of a symmetric and skew-symmetric matrix.

Key Concepts

Symmetric Matrix

A symmetric matrix is a special type of square matrix. It has the unique property that its transpose is equal to itself. In other words, if you swap the matrix's rows and columns, the matrix remains unchanged. This property can be mathematically expressed as

• Matrix \( \mathbf{B} \) is symmetric if \( \mathbf{B}^{T} = \mathbf{B} \).

Symmetric matrices are important in various fields of mathematics and physics because they often simplify problems due to their stable properties. Having symmetry means that, for such a matrix, the elements across the diagonal remain constant, i.e., element \( a_{ij} \) is equal to \( a_{ji} \). This occurs for all elements.

Some interesting facts about symmetric matrices include:

• All eigenvalues of a symmetric matrix are real numbers.
• Symmetric matrices are always diagonalizable.
• They occur naturally in contexts like second derivatives and covariance matrices.

Skew-Symmetric Matrix

A skew-symmetric matrix, also known as an antisymmetric matrix, presents another intriguing form of matrix. Unlike symmetric matrices, in a skew-symmetric matrix, the transpose is the negative of the original matrix. So, for a skew-symmetric matrix \(\mathbf{C} \)

• \( \mathbf{C}^{T} = -\mathbf{C} \).

A practical implication of this definition is that all elements on the main diagonal of a skew-symmetric matrix are zero. That's because for any element \(a_{ii}\), we must have \(a_{ii} = -a_{ii}\), which is only true if \(a_{ii} = 0\).

Some properties of skew-symmetric matrices include:

• The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
• In odd dimensions, skew-symmetric matrices have a determinant of zero.
• They often arise in physics, particularly in vector products and angular momentum.

Matrix Transpose

Matrix transpose is a basic yet very important operation that involves swapping the rows and columns of a matrix. For any given matrix \( \mathbf{A} \), its transpose is denoted as \( \mathbf{A}^{T} \).

• The element at the \( i^{th} \) row and \( j^{th} \) column of \( \mathbf{A} \) becomes the element at the \( j^{th} \) row and \( i^{th} \) column of \( \mathbf{A}^{T} \).

This operation is straightforward, yet it holds significance in many mathematical formulations and proofs.

Some important points to note about matrix transpose:

• Transposing a matrix twice will return you to the original matrix, \( \left(\mathbf{A}^{T}\right)^{T} = \mathbf{A} \).
• Transposition respects addition: \( \left(\mathbf{A} + \mathbf{B}\right)^{T} = \mathbf{A}^{T} + \mathbf{B}^{T} \).
• It also respects scalar multiplication: \( \left(k\mathbf{A}\right)^{T} = k\mathbf{A}^{T} \), for any scalar \( k \).

Understanding transpose helps in operations involving symmetric and skew-symmetric matrices. It allows us to establish their defining properties, ensuring matrices reach the desired form for specific algebraic operations.