Lecture 3: Linear systems Habib Ammari Department of Mathematics, ETH Z¨ urich Numerical methods for ODEs Habib Ammari
Linear systems • Linear systems: • Exponential of a matrix; • Linear systems with constant coefficients; • Linear system with non-constant real coefficients; • Second order linear equations; • Linearization and stability for autonomous systems. Numerical methods for ODEs Habib Ammari
Linear systems • Exponential of a matrix: • M d ( C ): vector space of d × d matrices with entries in C . • GL d ( C ) ⊂ M d ( C ): group of invertible matrices. • DEFINITION: Matrix norm � A � = max | y | =1 | Ay | . Numerical methods for ODEs Habib Ammari
Linear systems • LEMMA: Properties of the norm • | Ay | ≤ � A � | y | for all y ∈ C d ; • � A + B � ≤ � A � + � B � for all A , B ∈ M d ( C ); • � AB � ≤ � A � � B � for all A , B ∈ M d ( C ). Numerical methods for ODEs Habib Ammari
Linear systems • LEMMA: Jordan-Chevalley decomposition • A ∈ M d ( C ). • There exists C ∈ GL d ( C ) s.t. A has a unique decomposition C − 1 AC = D + N ; • D : Diagonal; N : Nilpotent (i.e., N d = 0); ND = DN . Numerical methods for ODEs Habib Ammari
Linear systems • Exponential of a matrix. • DEFINITION: • For A ∈ M d ( C ), � A n e A = n ! . n ≥ 0 Numerical methods for ODEs Habib Ammari
Linear systems • Properties: • Exponential of the sum: A , B ∈ M d ( C ), If AB = BA ⇒ e A + B = e A e B . • Conjugation and exponentiation: • A , B ∈ M d ( C ) and C ∈ GL d ( C ) s.t. A = C − 1 BC . • e A = C − 1 e B C . • PROOF: � � � A n ( C − 1 BC ) n C − 1 B n C e A = = C − 1 e B C ; n ! = = n ! n ! n ≥ 0 n ≥ 0 n ≥ 0 Numerical methods for ODEs Habib Ammari
Linear systems • Exponential of a diagonalizable matrix: • A : diagonalizable λ 1 0 ... A = C − 1 C ; 0 λ d • λ 1 , . . . , λ d ∈ C and C ∈ GL d ( C ). • ⇒ e λ 1 0 e A = C − 1 ... C . e λ d 0 Numerical methods for ODEs Habib Ammari
Linear systems • Exponential of a block matrix: • A j ∈ M h j ( C ) for j = 1 , ..., p ; A : block matrix of the form 0 A 1 ... A = . 0 A p • e A 1 0 e A = ... . e A p 0 Numerical methods for ODEs Habib Ammari
Linear systems • Derivative: A ∈ M d ( C ), d dt e tA = Ae tA = e tA A . Numerical methods for ODEs Habib Ammari
Linear systems • Linear systems with constant coefficients • A ∈ M d ( C )): independent of t . • f ∈ C 0 ([0 , T ]). • Linear ODE with constant coefficients: dx dt = Ax ( t ) + f ( t ) , t ∈ [0 , T ] , ( ∗ ) x (0) = x 0 ∈ R d . • for all x , y ∈ C d , | A ( x − y ) | ≤ � A �| x − y | • Cauchy-Lipschitz theorem ⇒ there exists a unique solution x to ( ∗ ). • ( ∗ ) autonomous system of equations. Numerical methods for ODEs Habib Ammari
Linear systems • If d = 1 (i.e., A = a ∈ C ), then by the method of integrating factors, � t x ( t ) = e at x 0 + e a ( t − s ) f ( s ) ds . 0 • General case ( d ≥ 1), if f = 0, x ( t ) = e tA x 0 . Numerical methods for ODEs Habib Ammari
Linear systems • For an arbitrary f , d dt ( e − tA x ) = e − tA f ( t ) , • ⇒ � t x ( t ) = e tA x 0 + e ( t − s ) A f ( s ) ds . 0 Numerical methods for ODEs Habib Ammari
Linear systems • Linear system with non-constant real coefficients • Homogeneous case; • Inhomogeneous case. • Homogeneous case: • M d ( R ): vector space of d × d matrices with entries in R . • PROPOSITION: • A : [0 , T ] → M d ( R ): continuous. • S : linear subspace of C 1 ([0 , T ]; R d ) of dimension d : � � x ∈ C 1 ([0 , T ]; R d ) : x satisfies dx S = dt = A ( t ) x Numerical methods for ODEs Habib Ammari
Linear systems • PROOF: • x , y ∈ S ⇒ for any α, β ∈ R , α x + β y ∈ C 1 ([0 , T ]; R d ): also a solution. • ⇒ S : linear subspace of C 1 ([0 , T ]; R d ). Numerical methods for ODEs Habib Ammari
Linear systems • Dimension of S = d : • Define F : S → R d by F [ x ] = x ( t 0 ) for some t 0 ∈ [0 , T ]. • F : linear: F [ α x + β y ] = α x ( t 0 ) + β y ( t 0 ) = α F [ x ] + β F [ y ] . • F : injective, F [ x ] = 0 ⇒ x = 0; • x solves dx dt = A ( t ) x ( t ) with the initial condition x ( t 0 ) = 0. • Cauchy-Lipschitz theorem ⇒ x = 0. Numerical methods for ODEs Habib Ammari
Linear systems • F : surjective: for any x 0 ∈ R d , dx dt = A ( t ) x ( t ) , t ∈ [0 , T ] , x ( t 0 ) = x 0 , has a solution x ∈ C 1 ([0 , T ]; R d ). Numerical methods for ODEs Habib Ammari
Linear systems • PROPOSITION: • x 1 , . . . , x d ∈ S ; • [ x 1 , . . . , x d ]: d × d matrix with columns x 1 , . . . , x d ∈ R d ; • det: determinant of a matrix; • Equivalent statements: (i) { x 1 , ..., x d } : basis of S ; (ii) det[ x 1 ( t ) , ..., x d ( t )] � = 0 for all t ∈ [0 , T ]. (iii) det[ x 1 ( t 0 ) , ..., x d ( t 0 )] � = 0 for some t 0 ∈ [0 , T ]. Numerical methods for ODEs Habib Ammari
Linear systems • PROOF: • (i) ⇔ (ii). • (i) ⇒ (iii): { x 1 , ..., x d } : basis of S ⇒ { F [ x 1 ] , ..., F [ x d ] } : basis of R d . • (iii) ⇒ (i): t 0 s.t. (iii) holds; F : S → R d : isomorphism relative to t 0 . • F − 1 : R d → S : isomorphism ⇒ x 1 = F − 1 [ x 1 ( t 0 )] , . . . , x d = F − 1 [ x d ( t 0 )]: basis of S . Numerical methods for ODEs Habib Ammari
Linear systems • DEFINITION: Fundamental matrix • If (i), (ii) or (iii): holds ⇒ x 1 , . . . , x d : fundamental system of solutions of the differential equation dx dt = A ( t ) x . • X = [ x 1 , . . . , x d ]: fundamental matrix of the equation. Numerical methods for ODEs Habib Ammari
Linear systems • DEFINITION: Wronskian determinant • x 1 , ..., x d ∈ S . • Wronskian determinant w ∈ C 1 ([0 , T ]; R ) of x 1 , . . . , x d : w ( t ) = det[ x 1 ( t ) , ..., x d ( t )] . Numerical methods for ODEs Habib Ammari
Linear systems • THEOREM: • x 1 , ..., x d ∈ S ; w ∈ C 1 ([0 , T ]; R d ): Wronskian determinant of x 1 , . . . , x d . • w solves the differential equation dw ( ∗∗ ) dt = ( tr A ( t )) w for t ∈ [0 , T ] . • t r : trace of a matrix. Numerical methods for ODEs Habib Ammari
Linear systems • PROOF: • If x 1 , ..., x d : linearly dependent → w = 0 and ( ∗∗ ) trivially holds. • Suppose that x 1 , ..., x d : linearly independent, i.e., w ( t ) � = 0 for all t ∈ [0 , T ]. • X : [0 , T ] → M d ( R ): fundamental matrix having as columns the solutions x 1 , ..., x d , i.e., X ( t ) = ( x ij ( t )) i , j =1 ,..., d , t ∈ [0 , T ] , x j = ( x 1 j , ..., x dj ) ⊤ for j = 1 , ..., d . Numerical methods for ODEs Habib Ammari
Linear systems • z j : solution of dz j dt = A ( t ) z j ( t ) , z j ( t 0 ) = e j , { e j } j =1 ,..., d : standard unit orthonormal basis in R d . Numerical methods for ODEs Habib Ammari
Linear systems • ⇒ { z 1 , . . . , z d } : basis of the space of solutions to dz / dt = Az . • There exists C ∈ GL d ( R d ) s.t. X ( t ) = CZ ( t ) , t ∈ [0 , T ] , Z = [ z 1 , . . . , z d ]. • v ( t ) := det Z ( t ) solves dv dt ( t 0 ) = t rA ( t 0 ) . • Z ( t 0 ) = I ⇒ v ( t 0 ) = 1. Numerical methods for ODEs Habib Ammari
Linear systems • Definition of the determinant of a matrix ⇒ � � d dv dt ( t ) = d ( − 1) sgn σ z i σ ( i ) ( t ) dt i =1 σ ∈ S d d � � � ( − 1) sgn σ d = dt z j σ ( j ) ( t ) z i σ ( i ) ( t ); σ ∈ S d j =1 i � = j S d : set of all permutations of the d elements { 1 , 2 , . . . , d } ; sgn σ : signature of the permutation σ . Numerical methods for ODEs Habib Ammari
Linear systems • � z i σ ( i ) ( t 0 ) = 0 unless σ = identity; i � = j • dz jj dt ( t 0 ) = ( A ( t 0 ) z j ( t 0 )) j � d � d = a jh ( t 0 ) z hj ( t 0 ) = a jh ( t 0 ) δ hj ( t 0 ) h =1 h =1 = a jj ( t 0 ) . • ⇒ d � dv dt ( t 0 ) = a jj ( t 0 ) = tr A ( t 0 ) . j =1 Numerical methods for ODEs Habib Ammari
Linear systems • Differentiation of w = det X = det( CZ ) = (det C ) det Z = (det C ) v ; • ⇒ dw dt ( t 0 ) = (det C ) dv dt ( t 0 ) = (det C )tr A ( t 0 ) . • v ( t 0 ) = 1 ⇒ dw dt ( t 0 ) = tr A ( t 0 ) w ( t 0 ) . Numerical methods for ODEs Habib Ammari
Linear systems • REMARK: • t 0 ∈ [0 , T ]. • Abel’s identity or Liouville’s formula: � t t 0 tr A ( s ) ds w ( t ) = w ( t 0 ) e for t ∈ [0 , T ] . • It suffices to check that the determinant of the fundamental matrix: nonzero for one t 0 ∈ [0 , T ]. Numerical methods for ODEs Habib Ammari
Linear systems • Inhomogeneous case • Inhomogeneous linear differential equation: � dx ( ∗ ∗ ∗ ) dt = A ( t ) x + f ( t ); • A ( t ) ∈ C 0 ([0 , T ]; M d ( R )) and f ∈ C 0 ([0 , T ]; R d ). • X : fundamental matrix for the homogeneous equation dx ( t ) / dt = A ( t ) x ( t ), dX dt = AX and det X � = 0 for all t ∈ [0 , T ] . • Any solution x to the homogeneous equation: x ( t ) = X ( t ) c , t ∈ [0 , T ] , for some (column) vector c ∈ R d . Numerical methods for ODEs Habib Ammari
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