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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) n cn 1n 1 c1 c0 • 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 of a linear tra nsformatio n T : V V if there is a nonzero vector x such that T (x) x. The vector x is called an eigenvecto r of T correspond ing to , and the setof all eigenvecto rs of (with the zero vector) is called the eigenspace 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 ' Pw ' y ' Ay APw 1 w ' P APw 7-21 • Quadratic Forms a' and c ' are eigenvalues of the matrix: matrix of the quadratic form 7-22 7-23 7-24