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Lecture 32.pdf

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MATH 136
Robert Sproule

Wednesday, March 26 − Lecture 32 : Eigenvalues of a matrix. Concepts: 1. Definition of eigenvalue of a matrix. 2. Definition of the characteristic polynomial of a square matrix A. 3. Find the eigenvalues of a matrix. 4. Recognize some important properties of characteristic polynomials of a matrix A: - The eigenvalues of a triangular matrix are the elements on its diagonal. - The number λ = 0 is an eigenvalue of A if and only if det(A) = 0. 1 T - The two matrices A and A have the same characteristic polynomial and hence the same eigenvalues. - Similar matrices have the same characteristic polynomial (C = B AB). -1 - The matrices AB and BA have the same characteristic polynomial. 32.1 Definitions − Let A be a square n × n matrix. Let λ be a variable over ℝ (the real numbers). Then det(A − λI) is a polynomial in λ. (Verify this for 2 × 2 and 3 × 3 matrices.) .The expression det(A − λI) will always turnout to be a polynomial in λ, of the form n -1 n det(A − λI) = a 0 a λ 1 ... + a n − 1 + λ . Then p(λ) = det(A − λI) is called the characteristic polynomial of the matrix A. The equation det(A − λI) = 0 is called the characteristic equation of A. n Note: Some authors define det(λI − A) = (−1) det(A − λI) as being the characteristic polynomial. Since this essentially produces the same characteristic equationdet(A − λI) = 0, we will not worry about this discrepancy. Any root of the characteristic polynomial of A is called an eigenvalue of A. That is, if λ 1 is a solution of the polynomial equation det(A − λI) = 0, then λ 1 is an eigenvalue of A. 32.1.1 Examples a) If A is the given 2 × 2 matrix, find all its eigenvalues. 2 det(A − λI) = (5 − λ) − 16 = (1 − λ)(9 −λ) = 0 The characteristic polynomial of Ais (1 − λ)(9 −λ) The characteristic equation of Ais (1 − λ)(9 −λ) = 0 Thus λ =11 and λ = 92are two only eigenvalues of the matrix A. Note that the order in which we present these eigenvalues is irrelevant. b) If A is the matrix then det(A − λI) = (2 − λ) [ (5 − λ) – 16] = (1 − λ)(2 − λ)(9 − λ) is characteristic polynomial of A and (1 − λ)(2 − λ)(9 − λ) = 0 is characteristic equation ofA. So 1, 2 and 9 are the only three eigenvalues of A. 3 32.1.2 Remark − If A is a matrix such that det(A − λI) = (4 − λ) (1 − λ), then we see that λ 1 4 is an eigenvalue and λ = 12is another eigenvalue. In this case we say that 4 is an eigenvalue of (algebraic) multiplicity 3. Definition – If an eigenvalue is a root of the characteristic polynomial of multiplicity m then we say it is an eigenvalue of algebraic multiplicity m. 32.1.3 Remark − If A is a matrix such that det(A − λI) = λ + 1, we see that A has no eigenvalues which are real numbers. We can however accept "complex" roots such as ±√(−1) = ± i. If A is an n by n matrix and we view det(A − λI) as a polynomial over the complex numbers ℂ, then there will always be n eigenvalues of A. Some of the eigenvalues may be identical. This follows from a fundamental theorem in algebra with says that any polynomial of degree n will have n complex roots. 32.1.4 Example − If A is the matrix we verify, by ex
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