A
square matrix M=m_ij is called
symmetric if
M^t = M (it is equal to its
transpose), or, equivalently,
m_ij = m_ji, or, equivalently, that for all
vectors v,w,
(Mv,w) = (v,Mw). This last property shows that a symmetric matrix stays symmetric in any
orthogonal basis! (
Note: Previously this last sentence was incorrectly phrased!)
n*n real symmetric matrices (symmetric matrices of real numbers) are important because they are guaranteed to be diagonalizable, i.e. they have n eigenvectors forming a base (or, equivalently, they have n eigenvalues if you count eigenvalues with their (geometric) multiplicity).