Science:Math Exam Resources/Courses/MATH307/April 2005/Question 01 (b)
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Question 01 (b) 

Consider the symmetric matrix (b) Find the number of positive and negative eigenvalues of B. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2005/Question 01 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. First we note that has all real entries and is symmetric so is Hermitian. It follows that the eigenvalues of must be real. With these conditions, we can apply the fact that the number of positive pivots = the number of positive eigenvalues which is given in by the following paper http://ocw.mit.edu/courses/mathematics/1806sclinearalgebrafall2011/positivedefinitematricesandapplications/symmetricmatricesandpositivedefiniteness/MIT18_06SCF11_Ses3.1sum.pdf Hence, since we found in part (a) that row operations yield
since there are 3 positive pivots, we must have 3 positive eigenvalues and 1 negative eigenvalue. We can also check this somewhat by considering that the determinant of a Hermitian matrix is the product of its eigenvalues. Recall that we found in part(a): The determinant of can be found by using part (a): . Note that we used the following determinant properties: For any matrices , we have For upper or lower triangular matrices, the determinant is the product of the diagonal entries Since we found that , there must be either 1 or 3 negative eigenvalues. This agrees with our earlier finding that there are 3 positive eigenvalues and 1 negative eigenvalue. 