of Matrix King Saud University August 29, 2018 of Matrix
Table of contents 1 2 of Matrix
Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that AX = λx. In this case, we say that X is an eigenvector of the matrix A with respect to the eigenvalue λ. of Matrix
Theorem If A M n (R) and λ R. λ is an eigenvalue the matrix A if and only if λi A = 0. of Matrix
Definition If A M n (R), the polynomial q A (λ) = λi A is called the characteristic equation of the matrix A. of Matrix
Example Find the eigenvalues of the following matrix 1 1 0 5 4 2 1 4 2 A = 1 2 1, A = 4 5 2, A = 3 4 0. 0 1 1 2 2 2 3 1 3 of Matrix
Theorem If A M n (R) and v 1,..., v m are eigenvectors for different eigenvalues λ 1,..., λ m, then v 1,..., v m are linearly independent. Proof We do the proof by induction. The result is true for m = 1. We assume the result for m and let v 1,..., v m+1 eigenvectors for different eigenvalues λ 1,..., λ m+1. of Matrix
If then Also we have Then a 1 v 1 +... a m v m + a m+1 v m+1 = 0 a 1 λ 1 v 1 +... a m λ m v m + a m+1 λ m+1 v m+1 = 0 a 1 λ m+1 v 1 +... a m λ m+1 v m + a m+1 λ m+1 v m+1 = 0 a 1 (λ 1 λ m+1 )v 1 +... + a m (λ m λ m+1 )v m = 0. Since (λ j λ m+1 0 for all j = 1,... m, then a 1 =... = a m = 0 and so a m+1 = 0. of Matrix
Definition We say that a matrix A M n (R) is diagonalizable if there exists an invertible matrix P M n (R) such that the matrix P 1 AP is diagonal. of Matrix
Remark If X 1,..., X n are the columns of the matrix P, then the columns of the matrix AP are: AX 1,..., AX n. Moreover if λ 1 0...... 0 0 λ 2 0.... D =.. 0............ 0 0...... 0 λ n then the columns of the matrix PD are: λ 1 X 1,..., λ n X n. Then P 1 AP = D PD = AP and the columns of the matrix P form a basis of R n and eigenvectors of the matrix A. of Matrix
Theorem The matrix A M n (R) is diagonalizable if and only if it has n eigenvectors linearly independent, then these vectors form a basis of the vector space R n. of Matrix
Examples Prove that the following matrices are diagonalizable and find an invertible matrix P M n (R) such that the matrix P 1 AP is diagonal and find A 15. 1 1 0 5 4 2 1 4 2 A = 1 2 1, A = 4 5 2, A = 3 4 0. 0 1 1 2 2 2 3 1 3 of Matrix
Definition Let A M n (R) and λ an eigenvalue of the matrix A. We define E λ = {X R n ; AX = λx } This space is called called the eigenspace associated to the eigenvalue λ. of Matrix
Remark If λ is an eigenvalue of the matrix A M n (R), then E λ = {X R n ; AX = λx } is vector sub-space of R n. Its dimension is called the the geometric multiplicity of λ. of Matrix
Definition If A M n (R) and the characteristic function q A (λ) = (λ λ 1 ) m Q(λ) such that Q(λ 1 ) 0 we say that m is the algebraic multiplicity of the eigenvalue λ 1. of Matrix
Theorem If A M n (R) and the characteristic function q A (λ) = C(λ λ 1 ) m 1... (λ λ p ) mp then A is diagonalizable if and only if the algebraic and geometric multiplicities are the same. of Matrix
Remark Special case If A M n (R) and has n different eigenvalues, then A is diagonalizable. of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. ( ) 5 4 A = 4 3 Solution The characteristic function of the matrix A is q A (λ) = 5 λ 4 4 3 λ = (1 λ)2. Then the matrix is not diagonalizable. of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. ( ) 10 6 A = 18 11 Solution The characteristic function of the matrix A is q A (λ) = 10 λ 6 18 11 λ = (λ 2)(1 + λ). Then the matrix is diagonalizable. of Matrix
E 1 = ( 2, 3) and E 2 = (1, ( 2). ) 1 0 The diagonal matrix is D = 0 2 ( ) 2 1 and the matrix P is P =. 3 2 of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. 5 0 4 A = 2 1 5 4 0 3 Solution The characteristic function of the matrix A is 5 λ 0 4 q A (λ) = 2 1 λ 5 4 0 3 λ = (1 λ)3. Then the matrix is not diagonalizable. of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. 1 0 0 A = 1 1 1 1 0 2 Solution The characteristic function of the matrix A is 1 λ 0 0 q A (λ) = 1 1 λ 1 1 0 2 λ = (1 λ)2 (2 λ). of Matrix
E 1 = (0, 1, 0), (1, 0, 1) and E 2 = (0, 1, 1). Then the matrix is diagonalizable. 1 0 0 the diagonal matrix is D = 0 1 0 0 0 2 0 1 0 and the matrix P is P = 1 0 1. 0 1 1 of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. 5 3 0 9 A = 0 3 1 2 0 0 2 0 0 0 0 2 Solution The characteristic function of the matrix A is 5 λ 3 0 9 q A (λ) = 0 3 λ 1 2 0 0 2 λ 0 = (5 λ)(3 λ)(2 λ) 2. 0 0 0 2 λ The matrix is diagonalizable if and only if the dimension of the vector space E 2 is 2. of Matrix
E 2 = (1, 1, 1, 0), ( 1, 2, 0, 1). Then the matrix A is diagonalizable. E 5 = (1, 0, 0, 0) and E 3 = (3, 2, 0, 0). 5 0 0 0 The diagonal matrix is D = 0 3 0 0 0 0 2 0 0 0 0 2 1 3 1 1 and the matrix P is P = 0 2 1 2 0 0 1 0 0 0 0 1. of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. 2 2 1 A = 1 3 1 1 2 2 Solution The characteristic function of the matrix A is 2 λ 2 1 q A (λ) = 1 3 λ 1 1 2 2 λ = (λ 1)2 (λ 5). of Matrix
E 1 = (1, 0, 1), ( 2, 1, 0), E 5 = (1, 1, 1). Then the matrix A is diagonalizable. 1 0 0 The diagonal matrix is D = 0 1 0 and the matrix P is P = 0 0 5 1 2 1 0 1 1 1 0 1 of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. 7 4 16 A = 2 5 8 2 2 5 Solution The characteristic function of the matrix A is 7 λ 4 16 q A (λ) = 2 5 λ 8 2 2 5 λ = (λ 3)2 (λ 1). E 3 = (1, 1, 0), (4, 0, 1), E 1 = (2, 1, 1). Then the matrix A is diagonalizable. of Matrix
1 0 0 The diagonal matrix is D = 0 3 0 and the matrix P is P = 0 0 3 2 1 4 1 1 0 1 0 1 of Matrix
Exercise Show if the following matrix is diagonalizable and find the matrix P such that the matrix P 1 AP is diagonal. A = 2 1 0 1 2 0 1 0 1 2 1 1 1 1 1 1 1 3 Solution The characteristic function of the matrix A is 1 2 λ 1 0 2 1 q A (λ) = 0 1 λ 0 2 1 1 1 λ 1 = (1 λ)(2 λ) 3. 1 1 1 3 λ The matrix is diagonalizable if and only if the dimension the vector space E 2 is 3. E 2 = ( 1, 1, 0, 2), Mongi ( 1, 0, BLEL 1, 0). of Matrix