Lecture 30 - Mar 20

Lecture 30 - Mar 20 - Friday March 20 Lecture 30...

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Friday March 20 Lecture 30 : Eigenvalues of a matrix. (Refers to section 6.1) Expectations: 1. Define and eigenvalue of a matrix. 2. Define 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 : o The eigenvalues of a triangular matrix are the elements on its diagonal. o The number λ 1 = 0 is an eigenvalue of A iff det( A ) = 0. o The two matrices A and A T have the same characteristic polynomial and hence the same eigenvalues. o Similar matrices have the same characteristic polynomial ( C = B -1 AB ). o The matrices AB and BA have the same characteristic polynomial. 30.1 Definitions Let A be a square n × n matrix. Let λ be a variable over R (the real numbers). Then det( A λ I ) is a polynomial in λ . (Verify this for 2 × 2 and 3 × 3 matrices.) .The expression det( A λ I ) will always turnout to be a polynomial in λ , of the form det( A λ I ) = a 0 + a 1 ( −λ ) + ... + a n 1 ( −λ ) n -1 + ( −λ ) n . Then p ( λ ) = det( A λ I ) is called the characteristic polynomial of the matrix A. The equation det( A λ I ) = 0 is called the characteristic equation of A . Note: Some authors define det( λ I A ) as being the characteristic polynomial. Since we are primarily interested in the characteristic equation and since the polynomial det( λ I A) = ( 1) n det(A λ I) essentially produces the same characteristic equation, we will not worry about this discrepancy .
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