根据已知条件有:A^T = A (A^T表示A的转置),A^2 = A * A = A^T * A=A.对任意的向量X,有 X^T * A * X = X^T * A^2 * X = X^T * A * A * X = X^T * A^T * A * X = (AX)^T * (AX),令AX = Y = (y1,...,yn), 则:X^T * A * X = X^T * A^2 * X = Y^T * Y = y1^2 + .+ yn^2 >= 0. 且 A 的行列式不为 0,根据 AX = Y,所以 X ≠ 0 => Y ≠ 0. 由 X 的任意性知道,A 为正定矩阵.