MAE101 ALG chapter 2 determinat and diagonalization

ĐẠI HỌC FPT CẦN THƠ Chapter Determinat & Diagonalization Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Contents ⚫ 3.1 The cofactor Expansion ⚫ 3.2.Determinants and Matrix Inverses ⚫ 3.3 Diagonalization and Eigenvalues ⚫ 3.4 An Application to Linear Recurrences Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 3.1 The Cofactor Expansion If A=[a] then the determinant of A, denoted by detA=a a ▪ If A is an 2x2 matrix then A =  ▪ det A = ▪ a b c d b  c d  = ad − bc + a A = d   g If A is an 3x3 matrix then the determinant of A is defined by a b c det A = d e f = +a g h i Dr Tran Quoc Duy e f h i −b d f g i +c d e g h - b e h + c f  i  Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ The determinant of 3x3 matrix a b c det A = d e f = +a g h i e f h i −b d f g i +c d e g h = aei + bfg + cdh − ceg − afh − bdi Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ The determinant of 3x3 matrix col1 col col col1 col a - b c a d e f d e g h i g h - - + col1 col col col1 col b + −5 −5 0 0 + det A = aei + bfg + cdh − ceg − afh − bdi Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Find detA if 1 −5  A = 0    0  col1 col col col1 col −5 −5 0 0 - + + + Note that : only use with 3x3 matrices Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ How to define the determinant of an mxm matrix? Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ The (i,j)-cofactor If A is an mxm matrix then the (i,j)-cofactor of A is defined by cij(A)=(-1)i+jdet(Aij) ⚫ Aij is the (m-1)x(m-1) matrix obtained from A by deleting row i and column j of A ⚫ For example, c23(A)=(-1)2+3det(A23)=-14 ⚫ 1 −1 0 A= 0 −1   Dr Tran Quoc Duy 5 3  0  2 1  A23 = 0    0  Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition If A is an mxm matrix then the determinant of A is defined by ⚫ detA=ai1ci1(A)+ai2ci2(A)+…+aimcim(A) ⚫ or detA= a1jc1j(A)+a2jc2j(A)+…+amjcmj(A) −1 0 = −1 = −68 −1 8 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Examples ⚫ Find the determinant of the matrices 1  A = 0    1  1 0 C= 0  0 −1  3  −1   0 2 Dr Tran Quoc Duy 0 0 B= 0  0 −1  3  −1   2 1 0 C'=  0  0 −1  3  0 2  −1  Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Characteristic Polynomial ⚫ ⚫ The characteristic polynomial of an nxn matrix A is defined by cA(x)=det(xI-A) 3  For example, if A =   1 −1   x − −5   c A ( x ) = det ( xI − A ) = det       −1 x + 1  = ( x − 3)( x + 1) − = x − x − Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Find the eigenvalues and eigenvectors of the matrix 1  A=  2 2 The characteristic polynomial of A is cA ( x) = x −1 −1 −2 x−2 = ( x − 1)( x − 2) − = x( x − 3) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ c A ( x) =  ( x − 1)( x − 2) − =  x( x − 3) = ▪ 1 = we solve the equation  −1 −1  −1 −1  −2 −2 →  0      The solution is  −1 t is arbitrary X =t  1 The eigenvectors of A corresponding to λ1=0 is  −1 X =t  1 Dr Tran Quoc Duy t≠0 Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ ▪ 2 = we solve the equation  −1  −1  −2  →  0      The solution is 1 X = t 2   1 t is arbitrary The eigenvectors of A corresponding to 2 = is 1 X = t 2   1 Dr Tran Quoc Duy t≠0 Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ When is A diagonalizable? Theorem A is diagonalizable iff every eigenvalue of multiplicity m yields exactly m basic eigenvectors, that is, iff the general solution of the system ( I − A) X = has exactly m parameters Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ 3.4 An Application to Linear Recurrences Dr Tran Quoc Duy Mathematics for Engineering Linear Dynamical ĐẠI System HỌC FPT CẦN THƠ Definition V0,V1,V2,… are columns such that V0 is known and Vk+1=AVk for each k>=0 matrix recurrence Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ An Application to Linear Recurrences Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ... FPT CẦN THƠ Example det A = −1 −1 1 −1 0 0 a)detA= 12 and det(A-1)T= 12 b) detA= 12 and det(A-1)T=1/ 12 c) detA=1/ 12 and det(A-1)T= 12 d) detA=1/ 12= det(A-1)T Dr Tran Quoc Duy Mathematics for Engineering... obtained from A by deleting row i and column j of A ⚫ For example, c23(A)=(-1 )2+ 3det(A23)=-14 ⚫ 1 −1 0 A= 0 −1   Dr Tran Quoc Duy 5 3  0  2 1  A23 = 0    0  Mathematics for... an mxm matrix then the determinant of A is defined by ⚫ detA=ai1ci1(A)+ai2ci2(A)+…+aimcim(A) ⚫ or detA= a1jc1j(A)+a2jc2j(A)+…+amjcmj(A) −1 0 = −1 = −68 −1 8 Dr Tran Quoc Duy Mathematics for Engineering