A real symmetric matrix is a real matrix that is also a symmetric matrix. Thus its entries are real numbers and it satisfies
 |
where 
denotes the transpose. Every real symmetric matrix
is a Hermitian matrix, since its conjugate
transpose equals its transpose.
By the spectral theorem, a real symmetric matrix has only real eigenvalues and an orthonormal basis of eigenvectors. Equivalently, it has a decomposition
 |
where 
is a real orthogonal matrix and 
is a real diagonal matrix
containing the eigenvalues. The columns of 
are the corresponding eigenvectors
(Olver 2010, pp. 13-14).
The real-entry condition matters. A complex symmetric matrix need not have real eigenvalues or be a
Hermitian matrix. For example, 
is symmetric but has eigenvalue 
, where 
is the imaginary unit and
is the identity matrix.
