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ECONOMETRICS phần 10

Dukes hoặc lãnh chúa của ngày trước đó. Tuy nhiên, sự khác biệt là trong thực tế rất lớn, cho một vị vua sô-cô-la không loại trừ tất cả, ông phục vụ. Anh ấy không cai trị lãnh thổ chinh phục, độc lập của thị trường, độc lập của khách hàng của mình. Sô-cô-la vua hay vua thép hoặc | APPENDIX A. MATRIX ALGEBRA 255 det AB det A det B det A I det A 1 A C B D det det D det A - BD C if det D 0 det A 0 if and only if A is nonsingular. If A is triangular upper or lower then det A Qk I aii If A is orthogonal then det A 1 A.7 Eigenvalues The characteristic equation of a square matrix A is det A - XIk 0. The left side is a polynomial of degree k in X so it has exactly k roots which are not necessarily distinct and may be real or complex. They are called the latent roots or characteristic roots or eigenvalues of A. If Xi is an eigenvalue of A then A XiIk is singular so there exists a non-zero vector hi such that A - XiIk hi 0. The vector hi is called a latent vector or characteristic vector or eigenvector of A corresponding to Xi . We now state some useful properties. Let Xi and hi i 1 . k denote the k eigenvalues and eigenvectors of a square matrix A. Let A be a diagonal matrix with the characteristic roots in the diagonal and let H hl hfc . det A nk i Xi tr A Pk i Xi A is non-singular if and only if all its characteristic roots are non-zero. If A has distinct characteristic roots there exists a nonsingular matrix P such that A P 1AP and PAP 1 A. If A is symmetric then A HAH0 and H0AH A and the characteristic roots are all real. A HAH0 is called the spectral decomposition of a matrix. The characteristic roots of A 1 are Xi X2 . Xk . The matrix H has the orthonormal properties H0H I and HH0 I . H 1 H0 and H 1 H A.8 Positive Definiteness We say that a k X k symmetric square matrix A is positive semi-definite if for all c 0 c0Ac 0. This is written as A 0. We say that A is positive definite if for all c 0 c0Ac 0. This is written as A 0. Some properties include APPENDIX A. MATRIX ALGEBRA 256 If A G G for some matrix G then A is positive semi-definite. For any c 0 c Ac a a 0 where a Gc. If G has full rank then A is positive definite. If A is positive definite then A is non-singular and A 1 exists. Furthermore A 1 0. A 0 if and only if it is symmetric and all