# Characteristic Polynomial - 國立臺灣大學 Characteristic Polyn omial Hung-yi Lee Outline Last lecture: Given eigenvalues, we know how to find eigenve ctors or eigenspaces Check eigenvalues This lecture: How to find eigenvalues? Reference: Textbook 5.2 Looking for Eigenvalues A scalar is an eigenvalue of A Existing such that = =0 Existing such that

( ) = 0 Existing such that has multiple solution The columns of are independent Dependent is not invertible Rank < n ( ) =0 Characteristic Polynomial A scalar is an eigenvalue of A ( ) =0 A is the standard matrix of linear operator T ( ) : Characteristic polynomial of A linear operator T

( ) =0 : Characteristic equation of A linear operator T Eigenvalues are the roots of characteristic polynomial or solutions of characteristic equation. Looking for Eigenvalues Example 1: Find the eigenvalues of A scalar is an eigenvalue of A ( ) =0 =0 t = -3 or 5 The eigenvalues of A are -3 or 5. Looking for Eigenvalues Example 1: Find the eigenvalues of

The eigenvalues of A are -3 or 5. Eigenspace of -3 =3 ( + 3 ) =0 find the solution Eigenspace of 5 =5 ( 5 ) =0 find the solution Looking for Eigenvalues Example 2: find the eigenvalues of linear operator standard matrix

( ) =0 A scalar is an eigenvalue of A 1 = 2 0 [ 0 1 0 0 1 1 ] ( ) =( 1 )

3 Looking for Eigenvalues Example 3: linear operator on R2 that rotates a vect or by 90 A scalar is an eigenvalue of A ( ) =0 standard matrix of the 90-rotation: No eigenvalues, no eigenvectors Characteristic Polynomial In general, a matrix A and RREF of A have different c haracteristic polynomials. Different Eigenvalues Similar matrices have the same characteristic polyn omials The same Eigenvalues

( ) ( 1 1 ( ) ) ( 1 ( ) = 1 ) ( 1 ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) Characteristic Polynomial Question: What is the order of the characteristic polynomial of an nn matrix A?

The characteristic polynomial of an nn matrix is indeed a polynomial with degree n Consider det(A tIn) Question: What is the number of eigenvalues of an nn mat rix A? Fact: An n x n matrix A have less than or equal to n eigen values Consider complex roots and multiple roots Characteristic Polynomial If nxn matrix A has n eigenvalues (including multiple roots) Sum of n eigenvalues Product of n eigenvalues = Trace of A

= Determinant of A Example Eigenvalues: -3, 5 Characteristic Polynomial The eigenvalues of an upper triangular matrix are it s diagonal entries. Characteristic Polynomial: [ 0 0

0 ] 0 0 [ 0

( ) ( )( ) The determinant of an upper triangular matrix is the product of its diagonal entries. ] Characteristic Polynomial v.s. Eigenspace Characteristic polynomial of A is ( ) Factorization 1 ( 1 ) 2

( 2 ) multiplicity ( ) ( ) Eigenvalue: 1 2 Eigenspace (dimension: )

1 2 1 2 Characteristic Polynomial v.s. Eigenspace Example 1: characteristic polynomials: (t + 1)2(t 3) Eigenvalue -1 Multiplicity of -1 is 2 Dim of eigenspace is 1 or 2

Dim = 2 Eigenvalue 3 Multiplicity of 3 is 1 Dim of eigenspace must be 1 Characteristic Polynomial v.s. Eigenspace Example 2: characteristic polynomials: (t + 1) (t 3)2 Eigenvalue -1 Multiplicity of -1 is 1 Dim of eigenspace must be 1 Eigenvalue 3 Multiplicity of 3 is 2 Dim of eigenspace is 1 or 2

Dim = 2 Characteristic Polynomial v.s. Eigenspace Example 3: characteristic polynomials: (t + 1) (t 3)2 Eigenvalue -1 Multiplicity of -1 is 1 Dim of eigenspace must be 1 Eigenvalue 3 Multiplicity of 3 is 2 Dim of eigenspace is 1 or 2 Dim = 1 Characteristic

polynomial Eigenvalues Eigenspaces -1 2 3 1 -1 1 3 2

-1 1 3 1 (t + 1)2(t 3) (t + 1) (t 3)2 (t + 1) (t 3)2 Summary Characteristic polynomial of A is ( ) Factorization 1 ( 1 )

2 ( 2 ) multiplicity ( ) ( ) Eigenvalue: 1 2 Eigenspace

(dimension: ) 1 2 1 2 Homework