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. In simpler terms, for a matrix A , if A[i][j] = A[j][i] for all indices i and j , then A is symmetric. Symmetric matrices are important in various fields, including mathematics, physics, and engineering. They often arise in problems involving linear algebra and can represent relationships in graphs or systems of equations. Their properties make them easier to analyze and compute, especially in optimization and eigenvalue problems.