1. What Is a Symmetric Matrix?
A symmetric matrix is a square matrix that remains the same when it is transposed. In other words,
\( A = A^T \)
This means every element mirrors perfectly across the main diagonal:
\( a_{ij} = a_{ji} \)
So the entry in row 1, column 3 must be equal to the entry in row 3, column 1, and so on.
2. Understanding the Structure of a Symmetric Matrix
For a matrix to be symmetric:
- It must be square (same number of rows and columns).
- The upper triangular part and the lower triangular part must be mirror images of each other.
- The diagonal can contain any numbers.
2.1. Example of a Symmetric Matrix
\( S = \begin{bmatrix} 2 & 3 & -1 \\ 3 & 5 & 4 \\ -1 & 4 & 7 \end{bmatrix} \)
Here:
- \( s_{12} = 3 \) and \( s_{21} = 3 \)
- \( s_{13} = -1 \) and \( s_{31} = -1 \)
- \( s_{23} = 4 \) and \( s_{32} = 4 \)
Everything matches across the diagonal, so the matrix is symmetric.
2.2. More Examples
- \( \begin{bmatrix} 4 & 1 \\ 1 & 9 \end{bmatrix} \) — a 2 × 2 symmetric matrix.
- \( \begin{bmatrix} 7 \end{bmatrix} \) — every 1 × 1 matrix is symmetric.
- \( \begin{bmatrix} 0 & -5 & 2 \\ -5 & 3 & 8 \\ 2 & 8 & 6 \end{bmatrix} \) — a 3 × 3 symmetric matrix.
3. How to Identify a Symmetric Matrix
To check if a matrix is symmetric, verify:
- Is it square?
- Does each entry satisfy \( a_{ij} = a_{ji} \)?
If every pair matches, the matrix is symmetric.
4. Key Observations About Symmetric Matrices
Symmetric matrices have some nice and useful properties:
- The diagonal elements can be any values.
- The matrix is completely determined by its upper (or lower) triangular part.
- Every scalar matrix and identity matrix is symmetric.
This makes symmetric matrices appear often in geometry, physics, statistics, and linear algebra applications.