A symmetric matrix is a square matrix in which the elements above and below the diagonal are equal. In other words, if the matrix is denoted as A, then A[i,j] = A[j,i] for all values of i and j.
A symmetric matrix is different from a regular matrix in that it has the special property of being equal to its own transpose. This means that if we were to flip the matrix over the main diagonal, the resulting matrix would be identical to the original.
Symmetric matrices are important in many areas of mathematics, particularly in linear algebra and matrix theory. They have useful properties that make them easier to work with and can be used to simplify complex mathematical problems.
In a symmetric matrix, definiteness refers to the nature of the eigenvalues of the matrix. A matrix with all positive eigenvalues is called positive definite, a matrix with all negative eigenvalues is called negative definite, and a matrix with both positive and negative eigenvalues is called indefinite.
The definiteness of a symmetric matrix can be determined by looking at the signs of the eigenvalues. If all eigenvalues are positive, the matrix is positive definite. If all eigenvalues are negative, the matrix is negative definite. If there are both positive and negative eigenvalues, the matrix is indefinite.