Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

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1 Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues of a square matrix A are the values of t that satisfy det(a ti n ). Proof v 0 such that (A - ti n )v = 0 det(a - ti n ) = 0.

2 Section 5. The Characteristic Polynomial Definition Example: A = apple 4 6 A ti = apple 4 t 6 t det(a ti )=( 4 t)(6 t) ( ) =(t + )(t 5). So, eigenvalues of A can be - or 5. Question: What can we find eigenvectors corresponding to -? What can we find eigenvectors corresponding to 5?

3 Example: A = apple 4 6 A ti = apple 4 t 6 t det(a ti )=( 4 t)(6 t) ( ) =(t + )(t 5). By solving ( A+ I) x = 0, weget a basis{[ 1] of A corresponding to the eigenvalue. By solving ( A 5I) x = 0, weget a basis{[ 1 of A corresponding to the eigenvalue 5. T ] T }of }of the eigenspace the eigenspace Question: Is the characteristic polynomial of A equal to its reduced row echelon form? Question: What is the order of the characteristic polynomial of an n n matrix A?

4 Properties: 1. In general, a matrix and its reduced row echelon form have different characteristic polynomials. Therefore, elementary operations are not useful in finding the characteristic polynomial of a matrix.. The characteristic polynomial of an n n matrix is indeed a polynomial with degree n, as can be deduced from the complete determinant expansion of det(a - ti n ).. The eigenvalues of an upper triangular matrix are its diagonal entries. Question: Now that elementary operations do not preserve characteristic polynomials, is there any other operations that do?

5 * Property Similar matrices have the same characteristic polynomials. Proof Suppose A and B are n n matrices that are similar. Assume B = P where P is invertible. 1 AP det(b ti n )= =det(a ti n )

6 * Property Similar matrices have the same characteristic polynomials. Proof

7 Definition Example: T : linear operator on R that rotates a vector by 90º. standard matrix is the 90º-rotation matrix characteristic polynomial of T is det apple apple ti =det t 1 1 t apple = t +1 Think: Consider an n x n matrix A. How many eigenvalues does it have? Can we predict the number of eigenvalues of A from its characteristic polynomial?

8 Fact: An n x n matrix A can have less than n eigenvalues due to the following reasons. 1) Complex (not real) solutions of its characteristic polynomial. ) Multiplicity of a real eigenvalue. Definition If is an eigenvalue of an n n matrix M, thenthelargest positive integer k such that (t ) k is a factor of the characteristic polynomial of M (i.e., det(m ti n )), is called the (algebraic) multiplicity of.

9 Example: Find the characteristic polynomials of A = B = and

10 The characteristic polynomials of A = B = and are -(t + 1) (t - ) and -(t + 1) (t - ), respectively, i.e., eigenvalue -1 eigenvalue -1 A: B: multiplicity 1 multiplicity 1 Then, find the eigenspaces corresponding to each of these eigenvalues.

11 The characteristic polynomials of A = B = and are -(t + 1) (t - ) and -(t + 1) (t - ), respectively, i.e., eigenvalue -1 eigenvalue -1 A: B: multiplicity 1 multiplicity 1 Then, find the eigenspaces corresponding to each of these eigenvalues.

12 The characteristic polynomials of A = B = and are -(t + 1) (t - ) and -(t + 1) (t - ), respectively, i.e., eigenvalue -1 eigenvalue -1 A: B: multiplicity 1 multiplicity 1 For A, the eigenspace corresponding to the eigenvalue has a basis 8 < : 1 9 = 5 ; dimension = 1 = multiplicity of the eigenvalue. For B, the eigenspace corresponding to the eigenvalue has a basis 8 < : 0 5, = 5 ; dimension = = multiplicity of the eigenvalue.

13 Facts & Questions: 1) The multiplicity of an eigenvalue λ of A is at least 1. ) When the multiplicity of λ is 1, then must the dimension of its eigenspace be 1? ) When the multiplicity of λ is k (where k > 1), then must the dimension of its eigenspace be k?

14 Example: Find The characteristic polynomial of C =

15 Example: The characteristic polynomial of is -(t + 1) (t - ), i.e., C: C = eigenvalue -1 multiplicity 1 Now, find the eigenspace corresponding to each of these eigenvalues. 5

16 Example: The characteristic polynomial of is -(t + 1) (t - ), i.e., eigenvalue -1 multiplicity 1 For C, the eigenspace corresponding to the eigenvalue has a basis C: C = dimension = 1 < multiplicity of the eigenvalue.

17 Definition Theorem 5.1

18 Theorem 5.1 *Proof Suppose A R n n, the geometric multiplicity of λ is k, and {v 1, v,, v k } is a basis of the eigenspace corresponding to λ. can extend {v 1, v,, v k } to a basis B = {v 1, v,, v n } of R n. Consider the linear operator T A on R n. For 1 j k, [T A (v j ] B =[Av j ] B =[ v j ] B = [v j ] B = e j B -matrix representation of T A has the form M = apple I k C O (n k) k D A and M are similar (Theorem 4.1 of Section 4.5). A and M have the same characteristic polynomial, which is

19 apple I det (M ti n ) = det k ti k C O D ti n k apple ( t)ik C = det O D ti n k = (det( t)i k )(det(d ti n k ) = ( t) k (det(d ti n k ) algebraic multiplicity of λ k. Example: T 4 x 1 x x 1 5A = 4 x 1 x 1 x x x standard matrix 5 A = characteristic polynomial det(a ti )=( 1 t)

20 Problem Set for Sections 5.1 and 5. Section 5.1: Problems 1,, 7, 9, 1, 15, 17, 19,, 5, 9, 1, 41, 4, 45, 47, 49, 51, 55, 57 Section 5.: Problems 1,, 5, 7, 9, 1, 15, 17, 1, 5, 7, 9,, 5, 9, 41, 5, 55, 57, 59, 61, 65, 69, 71.