Chapter 4

1. Chapter 4 Section 4.4 Eigenvalues and the Characteristic Polynomial

2. Characteristic Polynomial If A is an matrix the characteristic polynomial is a function of the variable t we call that is the determinant of . Characteristic Equation If is the characteristic polynomial of a matrix A, setting is the characteristic equation of the matrix A. The solutions to this equation are the eigenvalues of matrix A. Example For the matrix A given to the right find its characteristic polynomial and its eigenvalues. Since the 3rd row has two zeros we compute by expanding across the third row. Since the object of these problems is to find the roots of this polynomial we very often will leave the characteristic polynomial in factored form and do not multiply it out so we can tell what the solutions to are by just looking at it. Eigenvalues are: -1, 3, 4 Characteristic Equation:

3. Example Find the characteristic polynomial and the eigenvalues for the matrix A given to the right. (Sometimes called Fibonacci matrix.) Characteristic Polynomial is: This polynomial does not factor so use the quadratic formula to get If A is a matrix and is the characteristic polynomial for A then is a polynomial of degree nwith the coefficient of being . Using algebraic methods to solve a general polynomial of degree n can be difficult (in fact, sometimes impossible), but there are some techniques that will often work. Factoring and Solving Polynomials If is a polynomial with integer coefficients sometimes the following method will work for factoring the polynomial . Find the factors of the constant term say they are . Plug each factor into till you find one that is zero say (i.e. ), that means that is one of the factors. Use long division to find that means that . This gives one factor of . Apply the same thing to until you get down to degree 1 or 2 polynomial.

4. Example Find the characteristic polynomial and eigenvalues for the matrix A given to the right. Use a 1st column expansion then simplify the polynomial The constant term is 12, and the factors of 12 are: Plugging in we get Use long division to get Now we know Factor The characteristic polynomial is: 0 Eigenvalues are:

5. In general the problem of finding the characteristic polynomial and corresponding eigenvalues for a matrix can be very difficult. If the matrix has a certain pattern or contains a lot of zeros there are quick and easy ways to find the characteristic polynomial and eigenvalues. Characteristic Polynomials for Diagonal, Lower and Upper Triangular Matrices The characteristic polynomial of a diagonal, upper or lower triangular matrix is the product of each factor and the eigenvalues are the main diagonal entries. If diagonal or triangular: The Eigenvalues are: Example Find the characteristic polynomial and eigenvalues for the matrix A below to the right. The matrix is upper triangular. Eigenvalues: Example Find three different upper triangular matrices whose only eigenvalues are 1 and 4. Give all the possibilities for the characteristic polynomial Characteristic polynomial or

6. The number of times an eigenvalue is a root of the characteristic equation will tell us some information about the matrix when it is compared to another number that will be discussed later. This number is important enough we give it a name. Algebraic Multiplicity If A is a matrix and is the characteristic polynomial of the matrix A (i.e. ) and is an eigenvalue for A, then is a factor of the polynomial and the number of times that is a factor of is called the algebraic multiplicity of the eigenvalue . Algebraic multiplicity is the largest value such that: Example For the matrix to the right find the characteristic polynomial, all of the eigenvalues for the matrix along with the algebraic multiplicity of each of the eigenvalues that you found. The matrix A is diagonal so the characteristic polynomial is: The algebraic multiplicity of the eigenvalue 3 is equal to 4. The eigenvalues for A are : 3,7 The algebraic multiplicity of the eigenvalue 7 is equal to 2.

7. Example Find the characteristic polynomial and eigenvalues for the matrix A to the right. Even though A is not triangular take advantage of the zeros in the first two columns. 1st column expansion 2nd column expansion factor Characteristic polynomial , Eigenvalues: Properties of the Characteristic Polynomial We will now look at a few of the more important properties of the characteristic polynomial of a matrix A. If A is matrix and then: Characteristic Polynomials and Determinants The value of the characteristic polynomial at zero is the determinant of the matrix.

8. Evaluating Matrix Functions If you want to plug a matrix A into a polynomial (i.e. evaluate ) this is done by replacing all constant terms by in the polynomial where is the identity matrix. The rest of the polynomial is evaluated using ordinary matrix arithmetic. Being able to evaluate matrices in polynomials allows fro matrix functions trigonometric, exponential and logarithmic functions. Example For the matrix A to the right find the characteristic polynomial then evaluate . Characteristic Equation The result is the zero matrix ! Evaluating a Matrix in its Characteristic Polynomial Its is always true that if you plug a matrix into its characteristic polynomial you will get the zero matrix. If then: This result has many applications. One in particular is a more efficient way to compute the inverse of a matrix than augmenting with the identity and row reducing.

9. Example Find the characteristic polynomial of the matrix A to the right and use this to compute . Take advantage of the fact that the matrix A is upper triangular so : The idea here is to move the constant times identity matrix to one side of the equation. Factor the matrix A out from the other side. Divide by the constant if not 1. Collect all terms in factor other than A.

10. Eigenvalues and the Square of a Matrix If is an eigenvalue for the matrix A then is an eigenvalue of the matrix . If eigenvalues for A then the eigenvalues for are: The reason for this is as follows. Let v be the eigenvector corresponding to . Eigenvalues and a Power of a Matrix This is true for any positive integer p, if is an eigenvalue for the matrix A then is an eigenvalue of the matrix . If eigenvalues for A then the eigenvalues for are: The reason for this is as follows. Let v be the eigenvector corresponding to . If eigenvalues for A and is a polynomial then the eigenvalues for are: Eigenvalues and a Function of a Matrix If is an eigenvalue for the matrix A and is any polynomial then is an eigenvalue of the matrix . The reason for this is as follows. Let v be the eigenvector corresponding to . Let the polynomial .

11. Example A matrix A has eigenvalues of: . Find the eigenvalues of the matrices , and . The eigenvalues of are: The eigenvalues of are: Eigenvalues of are: If eigenvalues for A then the eigenvalues for are: Eigenvalues and the Inverse of a Matrix If is an eigenvalue for the matrix A then is an eigenvalue of the matrix . The reason for this is as follows. Let v be the eigenvector corresponding to . divide by If eigenvalues for A then the eigenvalues for are: The characteristic equation of and are equal. Eigenvalues and the Transpose of a Matrix The characteristic equation of a matrix A is equal to the characteristic equation of the matrix . This means a matrix and its transpose will have the same eigenvalues. The reason for this is as follows. Remember and .