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The Characteristic Equation of a Square Matrix (11/18/05) To this point we know how to check whether a given number is an eigenvalue of a given square matrix A, but we do not know how to find the eigenvalues of A (unless A is upper triangular). The trick is to use the determinant of A !

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Definition The characteristic equation of an n by n square matrix A is the equation det(A - I n ) = 0 This is a polynomial equation of degree n in the variable. Punch line: The roots of this equation are the eigenvalues of A.

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Two examples Find the eigenvalues of A = Find the eigenvalues of B =

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Why it works is an eigenvalue of a matrix A if and only if A x = x for some nonzero vector x, which is true if and only if (A - I ) x = 0 has a non-trivial solution, which is true if and only if det(A - I ) = 0. Is that cool or what?

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But there are drawbacks… First, determinants are not easy to compute, as we well know. Second, once the determinant is found, the polynomial equation may not be easy to solve. For example, what are the eigenvalues of matrices whose characteristic equations are: 1. 2 - 3 + 4 = 0 2. 3 - 2 2 + 5 - 3 = 0

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Assignment for Monday Finish up HW #3. Read Section 5.2. (Note: We already know the material on the determinant. They are assuming that the reader skipped Chapter 3) Do Exercises 1-11 odd, 15, and 19.

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