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Chapter 7 Eigenvalues and Eigenvectors 7.1 Eigenvalues and eigenvectors • Eigenvalue problem: If A is an nn matrix, do there exist nonzero vectors x in Rn such that Ax is a scalar multiple of x 7-1 7-2 • Note: Ax x ( I A) x 0 (homogeneous system) If ( I A ) x 0 has nonzero solutions iff det( I A ) 0. • Characteristic polynomial of AMnn: det( I A ) ( I A ) c n 1 n n 1 c1 c 0 • Characteristic equation of A: det( I A ) 0 7-3 7-4 • Notes: (1) If an eigenvalue 1 occurs as a multiple root (k times) for the characteristic polynomial, then 1 has multiplicity k. (2) The multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace. 7-5 7-6 • Eigenvalues and eigenvectors of linear transformations: A number is called an eigenvalue T : V V if there is a nonzero vector x such that The vector x is called an eigenvecto and the setof all eigenvecto called the eigenspace of a linear tra nsformatio n T ( x ) x. r of T correspond ing to , rs of (with the zero vector) is of . 7-7 7.2 Diagonalization • Diagonalization problem: For a square matrix A, does there exist an invertible matrix P such that P-1AP is diagonal? • Notes: (1) If there exists an invertible matrix P such that B P 1 AP , then two square matrices A and B are called similar. (2) The eigenvalue problem is related closely to the diagonalization problem. 7-8 7-9 7-10 7-11 7-12 7-13 7.3 Symmetric Matrices and Orthogonal 7-14 • Note: Theorem 7.7 is called the Real Spectral Theorem, and the set of eigenvalues of A is called the spectrum of A. 7-15 7-16 7-17 • Note: A matrix A is orthogonally diagonalizable if there exists an orthogonal matrix P such that P-1AP = D is diagonal. 7-18 7-19 7.4 Applications of Eigenvalues and Eigenvectors 7-20 • If A is not diagonal: -- Find P that diagonalizes A: y Pw y ' Pw ' w' P 1 P w ' y ' A y AP w AP w 7-21 • Quadratic Forms a ' and c ' are eigenvalues of the matrix: matrix of the quadratic form 7-22 7-23 7-24