A symmetric matrix is a square matrix that is equal to its transpose. This means that the elements are mirrored across the main diagonal, which runs from the top left to the bottom right. For example, in a symmetric matrix, the element in the first row and second column is the same as the element in the second row and first column. Symmetric matrices have important properties in various fields, including mathematics, physics, and engineering. They often arise in problems involving linear algebra and can be used to represent relationships in systems, such as in graph theory or optimization.