**Definitions:**

Assuming that a square matrix is positive definite and symmetric and the quadratic form of a given matrix from of the matrix equation is given. This form recasts the linear system () as the condition that the vector minimizes this quadratic form:

(1)

**Claim:**

The gradient of the function is equal to *iff* is a positive definite and symmetric matrix.

(2)

**Direct Proof:**

In this proof, both sides of the equation are fully expanded to derive equivalent statements given the specific properties of . For simplicity in the expansion we assume , this derivation can then be generalized to any .

(3)