Here is the theorem.


  1. The sum of two symmetric matrices is a symmetric matrix.
  2. If we multiply a symmetric matrix by a scalar, the result will be a symmetric matrix.
  3. If A and B are symmetric matrices then AB+BA is a symmetric matrix (thus symmetric matrices form a so-called Jordan algebra).
  4. Any power An of a symmetric matrix A (n is any positive integer) is a symmetric matrix.
  5. If A is an invertible symmetric matrix then A-1 is also symmetric.

Proof. 1 Let A and B be symmetric matrices of the same size. Consider A+B. We need to prove that A+B is symmetric. This means (A+B)T=A+B. Recall a property of transposes: the transpose of a sum is the sum of transposes. Thus (A+B)T=AT+BT. But A and B are symmetric. Thus AT=A and BT=B. So (A+B)T=A+B and the proof is complete.

Other parts of the theorem are left as an exercise.