线性代数chapter.ppt

Example ：The ellipse in Fig 7.4 (a) is the graph of the equation Find a change of variable that removes the cross-product term from the equation. Solution： The matrix of the quadratic form is . The eigenvalues of A turn out to be 3 and 7, with corresponding unit eigenvectors Let Then P orthogonally diagonally diagonalizes A, so the change of variable x=Py product the quadratic form . §7.2 Quadratic Forms 4. Classifying Quadratic Forms §7.2 Quadratic Forms Definition： A quadratic form Q is: a. positive definite if Q(x) 0 for all x ≠ 0. b. negative definite if Q(x) 0 for all x ≠ 0. c. indefinite if Q(x) assumes both positive and negative values. §7.2 Quadratic Forms Theorem 5 Quadratic Forms and Eigenvalues Let A be an n×n symmetric matrix. Then a quadratic form xTAx is: a. positive definite if and only if the eigenvalues of A are all positive. b. negative definite if and only if the eigenvalues of A are all negative c. indefinite if and only if A has both positive and negative eigenvalues. Proof ：By the Principal Axes Theorem, there exists an orthogonal change of variable x=Py such that where λ1, … , λn are the eigenvalues of A. Since P is invertible, there is a one-to-one correspondence between all nonzero x and all nonzero y. Thus the values of Q(x) for x≠0 coincide with the values of the expression on the right side of (4), with is obviously controlled by the signs of the eigenvalues λ1, … , λn , in the three ways described in the theorem. §7.2 Quadratic Forms Example ： Is Q(x) positive definite? Solution： Because of all the plus signs, the form “looks” positive definite. But the matrix of the form is and the eigenvalues of A turn out to be 5,2, and -1. So Q is an indefinite quadratic. CHAPTER 7Symmetric Matrices and Quadratic Forms Chapter 7 Symmetric Matrices and Quadratic Forms § 7.1 D