研究生的筆記本

Definition:

A symmetric matrix is matrix that equals its own transpose.

i.e.

Remarks:

1. A symmetric matrix must be a square matrix.

2. A symmetric matrix can be a singular matrix.

(It is possible that a symmetric matrix is not invertible.)

For example, 

is symmetric

is symmetric

is not symmetric

3. If a symmetric matrix is invertible, then its inverse must be symmetric.

proof:

4. If A is symmetric and if it can be factored as LDU, then

proof:

Note that LDU factorization is unique. 

U: upper triangular matrix; L: lower triangular matrix.