Symmetric matrix

A symmetric matrix is a matrix whose transpose is equal to . All real-valued diagonal matrices are symmetric. Symmetric matrices have some interesting properties:

• The sum of two symmetric matrices is also symmetric.
• For any integer , is symmetric if is symmetric.
• The rank of a symmetric matrix is equal to the number of non-zero eigenvalues of .
• The eigenvectors of a symmetric matrix corresponding to distinct eigenvalues are orthogonal.
  • For repeated eigenvalues, the eigenvectors can be chosen to form an orthonormal set.

Perhaps most importantly, every symmetric matrix can be diagonalized by an orthogonal matrix. This is called the spectral theorem.