Symmetric Matrix
A symmetric matrix is a square matrix that is equal to its transpose. Mathematically, a matrix \( A \) is symmetric if:
\[ A = A^T \]
where \( A^T \) is the transpose of \( A \), meaning that the rows and columns of \( A \) are interchanged.
Examples of Symmetric Matrices
Example 1: A 2×2 Symmetric Matrix
Consider the matrix:
\[ A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix} \]
Its transpose is:
\[ A^T = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix} \]
Since \( A = A^T \), the matrix is symmetric.
Example 2: A 3×3 Symmetric Matrix
\[ B = \begin{bmatrix} 5 & -1 & 3 \\ -1 & 2 & 4 \\ 3 & 4 & 1 \end{bmatrix} \]
Its transpose:
\[ B^T = \begin{bmatrix} 5 & -1 & 3 \\ -1 & 2 & 4 \\ 3 & 4 & 1 \end{bmatrix} \]
Since \( B = B^T \), it is symmetric.
Properties of Symmetric Matrices
Property 1: The Sum of Two Symmetric Matrices is Symmetric
If \( A \) and \( B \) are symmetric matrices of the same size, then their sum \( A + B \) is also symmetric.
Example
Let:
\[ A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 \\ 5 & 6 \end{bmatrix} \]
Then:
\[ A + B = \begin{bmatrix} 1+4 & 2+5 \\ 2+5 & 3+6 \end{bmatrix} = \begin{bmatrix} 5 & 7 \\ 7 & 9 \end{bmatrix} \]
Since \( A + B = (A + B)^T \), the result is symmetric.
Property 2: The Scalar Multiple of a Symmetric Matrix is Symmetric
If \( A \) is a symmetric matrix and \( c \) is a scalar, then \( cA \) is also symmetric.
Example
Let:
\[ A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \quad c = 3 \]
Then:
\[ 3A = \begin{bmatrix} 6 & 9 \\ 9 & 15 \end{bmatrix} \]
Since \( (3A)^T = 3A \), it remains symmetric.
Property 3: The Product of Two Symmetric Matrices is Symmetric if They Commute
If \( A \) and \( B \) are symmetric matrices and \( AB = BA \), then \( AB \) is also symmetric.
Example
Let:
\[ A = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 2 \\ 2 & 5 \end{bmatrix} \]
Calculate:
\[ AB = \begin{bmatrix} 2(4) + (-1)(2) & 2(2) + (-1)(5) \\ -1(4) + 3(2) & -1(2) + 3(5) \end{bmatrix} = \begin{bmatrix} 8 – 2 & 4 – 5 \\ -4 + 6 & -2 + 15 \end{bmatrix} \]
\[ = \begin{bmatrix} 6 & -1 \\ -1 & 13 \end{bmatrix} \]
Since \( AB = AB^T \), the product is symmetric.
Property 4: All Eigenvalues of a Symmetric Matrix are Real
If \( A \) is a symmetric matrix, then all its eigenvalues are real numbers.
Example
Consider the symmetric matrix:
\[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \]
Finding its eigenvalues involves solving:
\[ \det(A – \lambda I) = 0 \]
\[ \begin{vmatrix} 2 – \lambda & -1 \\ -1 & 2 – \lambda \end{vmatrix} = 0 \]
\[ (2 – \lambda)(2 – \lambda) – (-1)(-1) = 0 \]
\[ \lambda^2 – 4\lambda + 3 = 0 \]
Solving for \( \lambda \):
\[ \lambda = \frac{4 \pm \sqrt{16 – 12}}{2} = \frac{4 \pm 2}{2} \]
\[ \lambda = 3, 1 \]
Both eigenvalues are real.
Property 5: A Symmetric Matrix is Diagonalizable
Every symmetric matrix is diagonalizable, meaning it can be expressed as:
\[ A = PDP^{-1} \]
where \( P \) is an orthogonal matrix and \( D \) is a diagonal matrix containing eigenvalues of \( A \).
Example
For:
\[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \]
The eigenvalues are \( \lambda = 3, 1 \) (as found earlier). The corresponding eigenvectors form the columns of \( P \), and the diagonal matrix \( D \) consists of the eigenvalues:
\[ D = \begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix} \]