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