tailieunhanh - EBook - Mathematical Methods for Robotics and Vision Part 8

Tham khảo tài liệu 'ebook - mathematical methods for robotics and vision part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 64 CHAPTER 5. EIGENVALUESAND EIGENVECTORS Eigenvalues Vectors and Singular Values Vectors In this section we prove a few additional important properties of eigenvalues and eigenvectors. In the process we also establish a link between singular values vectors and eigenvalues vectors. While this link is very important it is useful to remember that eigenvalues vectors and singular values vectors are conceptually and factually very distinct entities recall figure . First a general relation between determinant and eigenvalues. Theorem The determinant of a matrix is equal to the product of its eigenvalues. Proof. The proof is very simple given the Schur decomposition. In fact we know that the eigenvalues of a matrix are equal to those of the triangular matrix in the Schur decomposition of . Furthermore we know from theorem that the determinant of a triangular matrix is the product of the elements on its diagonal. If we recall that a unitary matrix has determinant 1 or -1 that the determinants of and are the same and that the determinant of a product of matrices is equal to the product of the determinants the proof is complete. We saw that an X Hermitian matrix with distinct eigenvalues admits orthonormal eigenvectors corollary . The assumption of distinct eigenvalues made the proof simple but is otherwise unnecessary. In fact now that we have the Schur decomposition we can state the following stronger result. Theorem Spectral theorem Every Hermitian matrix can be diagonalized by a unitary matrix and every real symmetric matrix can be diagonalized by an orthogonal matrix real real In either case is real and diagonal. Proof. We already know that Hermitian matrices and therefore real and symmetric ones have real eigenvalues theorem so is real. Let now be the Schur decomposition of . Since is Hermitian so is . In fact and But the only way that can be both triangular and Hermitian is for it to be diagonal because . Thus the Schur .