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Unformatted text preview: Symmetric matrices and the second derivative test 1 Chapter 4 Symmetric matrices and the second derivative test In this chapter we are going to finish our description of the nature of nondegenerate critical points. But first we need to discuss some fascinating and important features of square matrices. A. Eigenvalues and eigenvectors Suppose that A = ( a ij ) is a fixed n n matrix. We are going to discuss linear equations of the form Ax = x, where x R n and R . (We sometimes will allow x C n and C .) Of course, x = 0 is always a solution of this equation, but not an interesting one. We say x is a nontrivial solution if it satisfies the equation and x 6 = 0. DEFINITION. If Ax = x and x 6 = 0, we say that is an eigenvalue of A and that the vector x is an eigenvector of A corresponding to . EXAMPLE. Let A = 0 3 1 2 . Then we notice that A 1 1 = 3 3 = 3 1 1 , so 1 1 is an eigenvector corresponding to the eigenvalue 3. Also, A 3- 1 =- 3 1 =- 3- 1 , so 3- 1 is an eigenvector corresponding to the eigenvalue- 1. EXAMPLE. Let A = 2 1 0 0 . Then A 1 = 2 1 , so 2 is an eigenvalue, and A 1- 2 = , so 0 is also an eigenvalue. REMARK. The German word for eigenvalue is eigenwert . A literal translation into English would be characteristic value, and this phrase appears in a few texts. The English word eigenvalue is clearly a sort of half translation, half transliteration, but this hybrid has stuck. 2 Chapter 4 PROBLEM 41. Show that A is invertible 0 is not an eigenvalue of A . The equation Ax = x can be rewritten as Ax = Ix , and then as ( A- I ) x = 0. In order that this equation have a nonzero x as a solution, Problem 352 shows that it is necessary and sufficient that det( A- I ) = 0 . (Otherwise Cramers rule yields x = 0.) This equation is quite interesting. The quantity det a 11- a 12 ... a 1 n a 21 a 22- ... a 2 n . . . a n 1 a n 2 ... a nn- can in principle be written out in detail, and it is then seen that it is a polynomial in of degree n . This polynomial is called the characteristic polynomial of A ; perhaps it would be more consistent to call it the eigen polynomial, but no one seems to do this. The only term in the expansion of the determinant which contains n factors involving is the product ( a 11- )( a 22- ) ... ( a nn- ) . Thus the coefficient of n in the characteristic polynomial is (- 1) n . In fact, that product is also the only term which contains as many as n- 1 factors involving , so the coefficient of n- 1 is (- 1) n- 1 ( a 11 + a 22 + + a nn ). This introduces us to an important number associated with the matrix A , called the trace of A : trace A = a 11 + a 22 + + a nn ....
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