Linear Transformations x x = y j= b = 3 4 0 1 i = 1 0 Ax = b x = A-1 b Linear Transform A, transform a vector into another vector, i. e change its magnitude & direction
2차 형식(Quadratic Form) f = ax 2 + 2 bxy + cy 2 = x y a b b c x y f = x. TAx = x 1 x 2 ……. xn 임의의 대칭행렬 A에 대해 f = x. TAx는 2차 형식이 된다. a 11 a 21 …. an 1 …. … a 1 n a 2 n …. ann x 1 x 2. . . xn = a 11 x 12 + a 12 x 1 x 2 + a 21 x 2 x 1 + …. + annx 2 n = ΣΣ aijxixj
Spectral Theorem x= AQ = QΛ x. TAx = x. T q 1 q 2 QTAQ = Λ q 1 q 2 qn = qn y. T (QTx)T= y. T x. TQ = y. T x. TAx = y. TΛy A = QΛQT q 1 T q 2 T. q n. T λ 1 0. . 0 0 λ 2. . 0 ……… 0 0. . λn q 1 T q 2 T. q n. T x = y QTx = y x