eigen value eigen vector Linear Transformations x x

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고유치와 고유벡터 (eigen value & eigen vector)

고유치와 고유벡터 (eigen value & eigen vector)

Linear Transformations x x = y j= b = 3 4 0 1 i

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

Trace of Matrix A= 1 0. 9 0. 2 0. 9 1 0. 5

Trace of Matrix A= 1 0. 9 0. 2 0. 9 1 0. 5 0. 2 0. 5 1 tr(A) = Σaii 1) tr(c. A) = c. tr(A) 2) tr(A±B) = tr(A) ± tr(B) 3) tr(AB) = tr(BA) tr(AB) = ΣΣaijbji, tr(BA) = Σσbikajk 4) tr(ABC) = tr(CAB) = tr(BCA) tr(A(BC)) = tr((BC)A) 5) tr(B-1 AB) = tr(A)

2차 형식(Quadratic Form) f = ax 2 + 2 bxy + cy 2 =

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

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

Spectral Theorem AS = SΛ 상관행렬 1 0. 9 0. 2 A = SΛS-1

Spectral Theorem AS = SΛ 상관행렬 1 0. 9 0. 2 A = SΛS-1 λ 1 = 2. 12, λ 2 = 0. 83, λ 3 = 0. 05 0. 9 1 0. 5 S-1 AS = Λ 0. 2 0. 5 1 = 0. 61 q 1= 0. 68 0. 41 -0. 47 q 2= -0. 11 0. 88 0. 61 -0. 47 0. 64 0. 68 -0. 11 -0. 73 0. 41 0. 88 0. 25 0. 64 q 3= -0. 73 0. 25 2. 12 0. 83 0. 05 0. 61 0. 68 0. 41 -0. 47 -0. 11 0. 88 0. 64 -0. 73 0. 25