Systems of First Order Linear Equations Ordinary Differential Equations Dr. Marco A Roque Sol 12/01/2015 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Example 7.7 Solve the system of equations x1 − 2x2 + 3x3 = b1 −x1 + x2 − 2x3 = b2 2x1 − x2 − 3x3 = b3 for various values of b1 , b2 , and b3 Solution By performing steps (a), (b), and (c) as in Example 7.6, we transform the matrix into Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 1 −2 0 0 1 0 b1 −1 −b1 − b2 0 b1 + 3b2 + b3 3 The equation corresponding to the third row is b1 + 3b2 + b3 = 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors thus the system has no solution unless the above condition is satisfied by b1 , b2 , and b3 . b1 = −3b2 − b3 Assuming that the condition is satisfied 1 −2 3 −3b2 − b3 −(−3b2 − b3 ) − b2 0 1 −1 0 0 0 (−3b2 − b3 ) + 3b2 + b3 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 1 −2 3 −3b2 − b3 0 1 −1 2b2 + b3 0 0 0 0 Add (2) times the second row to the first row. 1 0 1 −3b2 − b3 b2 + b3 0 1 −1 0 0 0 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Thus, we have two equations and one unknown, so one of the variables let’s say x3 , is equal to a parameter α, obtaining the system x1 + α = −3b2 − b3 x2 − α = b2 + b3 Hence, we obtain x1 = −α − 3b2 − b3 ; x2 = α + b2 + b3 −α − 3b2 − b3 −1 −3b2 − b3 α + b2 + b3 = α 1 + b2 + b3 X= α 1 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Linear Dependence and Independence . A set of k vectors x(1) , ..., x(k) is said to be linearly dependent if there exists a set of real or complex numbers c1 , ...., ck , at least one of which is nonzero, such that c1 x(1) + ... + ck x(k) = 0 On the other hand, if the only set c1 , ..., ck for which the above equation is satisfied is c1 = c2 = · · · = ck = 0,then the set of vectors x(1) , ..., x(k) is called linearly independent. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Consider now a set of n vectors, each of which has n components , x1n x12 x11 x2n x22 x21 x(1) = . ; x(2) = . ; · · · x(n) = . .. .. .. xnn xn2 xn1 the above equation can be written as . x11 c1 + x12 c2 + . . . + x1n cn x21 c1 + x22 c2 + . . . + x2n cn =0 .. .. . . xn1 c1 + xn2 c2 + . . . + xnn cn Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors or equivalently Xc = 0 (X−1 If X is nonsingular exists), then the only solution of is c = 0, −1 but if X is singular (X does not exist) there are nonzero solutions. Example 7.8 Determine wether the vectors are linearly indepent or not 1 2 − 4 2 ; x(2) = 1 ; x(3) = 1 x(1) = − 1 3 − 11 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Solution To determine whether x (1) , x (2) , and x (3) are linearly dependent, we seek constants c1 , c2 , and c3 such that c1 x(1) + c2 x(2) + c3 x(3) = 0 written in the matrix form 1 2 4 c1 2 1 1 c2 = 0 −1 3 −11 c3 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Using elementary row operations on the 1 2 −4 1 2 1 −1 3 −11 augmented matrix 0 0 0 (a) Add (−2) times the first row to the second row, and add the first row to the third row. 1 2 −4 0 0 −3 9 0 0 5 −15 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (b) Divide the second row by (−3), second row to the third row. 1 2 −4 0 1 −3 0 0 0 then add (−5) times the 0 0 0 Thus we obtain the equivalent system c1 + 2c2 − 4c3 = 0 =0 c2 − 3c3 = 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Hence, we have 2 equations in 3 unknowns, so one of them, let’s say c3 will be a free parameter (real number) α and the solution of the system is c3 = α; c2 = 3c3 = 3α; c1 = −2c2 + 4c3 = −2c3 = −2α c1 − 2α − 2 c2 = 3α = α 3 c3 α 1 Hence, there are infinitely solutions and the set of vectors is linearly dependent. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Determinants Associated to every n × n matrix A there is a real number called the determinant of A denoted by |A| or det(A) and defined inductivly as follows n = 1 A = a11 a11 a12 n=2 A= a21 a22 |A| = a11 a11 a12 = a11 a22 − a12 a21 |A| = a21 a22 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors a11 a12 a13 n = 3 A = a21 a22 a23 a31 a32 a33 a11 a12 a13 |A| = a21 a22 a23 = a31 a32 a33 a22 a23 a21 a23 a11 a12 − a12 a11 a31 a33 + a13 a31 a32 = a11 a22 a33 + a32 a33 a12 a23 a31 + a13 a21 a32 − a31 a22 a13 − a32 a23 a11 − a33 a21 a12 = Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31 a21 a22 a23 a21 a22 +a13 a21 a32 − a31 a22 a13 = a31 a32 a33 a31 a32 −a32 a23 a11 − a33 a21 a12 − − − + + + Now, for n ≥ 4, if let M1j be the corresponding minors to the first row, then we have n X |A| = (−1)1+j a1j M1j j=1 or using any fix row i |A| = n X j=1 Dr. Marco A Roque Sol (−1)i+j aij Mij Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Theorem 7.1 Given an n × n matrix A, if B is an n × n matrix obtained from A by 1) Adding a multiple of the ith row (column) to the jth row then |B| = |A| 2) Interchanging two consecutive rows (columns), then |B| = −|A| 3) Multiplying a row (column) by a nonzero scalar α then |B| = α|A| Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Theorem 7.2 1) If A has a row (column) of zeros, then |A| = 0 2) If A has a two identical rows (columns), then |A| = 0 3) If two rows (columns) of A are proportional, then |A| = 0 4) If A ia upper (lower) triangular matrix, then |A| = a11 a22 a33 · · · ann Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 5) |AT | = |A| 6) |AB| = |A||B| Example 7.9 Find the following determinant 1 −1 A= 0 −3 of the matrix Dr. Marco A Roque Sol −1 2 4 3 −2 1 2 1 0 1 1 −1 Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Solution 1 −1 2 4 −1 3 −2 1 = |A| = 2 1 0 0 −3 1 1 −1 1 −1 2 4 0 2 0 5 0 0 1 −5 = 0 0 7 16 1 −1 2 4 0 2 0 5 0 2 1 0 = 0 −2 7 11 1 −1 2 4 0 2 0 5 0 0 1 −5 = (1)(2)(1)(51) = 102 0 0 0 51 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Theorem 7.3 A matrix A is singular ⇐⇒ |A| = 0 ⇐⇒ Ax = 0 has a nonzero solution ⇐⇒ Columns of A are linearly dependent. Eigenvalues and Eigenvectors. The equation Ax = y can be viewed as a linear transformation that maps (or transforms) a given vector x into a new vector x. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Given an n × n matrix A we consider the problem of finding a vector x that is transformed into a multiple of itself Ax = λx but this is equivalent to say that Ax = λIx Ax − λIx = 0 (A − λI) x = 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The latter equation has nonzero solutions if and only if λ is chosen so that |A − λI| = 0 This is a polynomial equation of degree n in λ and is called the characteristic equation of the matrix A Values of λ may be either real- or complex-valued and are called eigenvalues of A . The nonzero vectors that are obtained by using such a value of λ are called the eigenvectors corresponding to that eigenvalue. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors a) It is possible to show that if λ1 and λ2 are two eigenvalues of A and if λ1 6= λ2 , then their corresponding eigenvectors x (1) and x (2) are linearly independent. This result extends to any set λ1 , ..., λk of distinct eigenvalues: their eigenvectors x (1) , ..., x (k) are linearly independent. Thus, if each eigenvalue of an n × n matrix is simple, then the n eigenvectors of A , one for each eigenvalue, are linearly independent. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors b) On the other hand, if A has one or more repeated eigenvalues, then there may be fewer than n linearly independent eigenvectors associated with A, since for a repeated eigenvalue with multiplicity m, we may have q < m linearly independent vectors. c) In the case of an eigenvalue, λi with multiplicity m, if we can find m eigenvectors x (i1) , x (i2) , ..., x (im) , linearly independent associated to λi , we say that the matrix is Non-defective. d) Otherwise, if we are able to find just x (i1) , x (i2) , ..., x (iq) ; q < m linearly independent associated to λi , we say that the matrix is Defective. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Example 7.10 Find the eigenvalues and eigenvectors 0 1 A = 1 0 1 1 of the matrix 1 1 0 Solution The eigenvalues λ and eigenvectors x satisfy the equation (A − λI) x = 0, or Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors −λ 1 1 x1 0 1 −λ 1 x2 = 0 |A − λI| = 1 1 −λ x3 0 The eigenvalues are the roots of the equation −λ 1 1 −λ 1 1 −λ 1 1 1 = 1 1 −λ = |A − λI| = 1 −λ 1 = − −λ 1 1 1 1 −λ 1 −λ −λ 1 1 1 −λ 1 0 λ+1 −1 − λ = −λ3 + 3λ2 + 2 = 0 0 −λ2 + 1 λ + 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 . 1) For λ1 = 2 −2 1 1 x1 0 1 −2 1 x2 = 0 1 1 −2 x3 0 We can reduce this to 2 0 0 the equivalent system −1 −1 x1 0 1 −1 x 2 = 0 0 0 x3 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors by elementary row operations. Solving this system yields the eigenvector 1 (1) x = 1 1 2) For λ2 = −1 1 1 1 x1 0 1 1 1 x2 = 0 1 1 1 x3 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The above system is reduced immediately to the single equation x1 + x2 + x3 = 0 One equation and three unknowns. Hence, two of them are free veriables, let’s say x1 = c1 , x2 = c2 , and x3 = −c1 − c2 . Thus we have c1 1 0 c2 x= = c1 0 + c2 1 −c1 − c2 −1 −1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors In this way two linearly independent eigenvectors associated to λ2 = −1 are 1 0 0 x(3) = 1 x(2) = − 1 − 1 Thus, the three linearly independent eigenvectors, are 1 1 0 0 x(3) = 1 x(1) = 1 x(2) = 1 − 1 − 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Example 7.11 Find the eigenvalues and eigenvectors of the matrix 2 −3 −1 A = 0 −1 0 −1 1 2 Solution The eigenvalues λ and eigenvectors x satisfy the equation (A − λI) x = 0, or Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 2−λ −3 −1 x1 0 0 −1 − λ 0 x2 = 0 |A − λI| = −1 1 2−λ x3 0 The eigenvalues are the roots of the equation 2 − λ 2 − λ −3 −1 −3 −1 −1 − λ 0 = − −1 1 2 − λ = |A − λI| = 0 −1 1 2−λ 0 −1 − λ 0 −1 1 2 − λ −1 1 2 − λ 2 − λ −3 −1 = 0 −1 − λ (2 − λ)2 − 1 = 0 −1 − λ 0 0 −1 − λ 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors −1 2 − λ 1 2 − 0 (2 − λ) − 1 −1 − λ = (1 + λ) (2 − λ)2 − 1 = 0 0 0 −1 − λ The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 . 1) For λ1 = −1 2−λ −3 −1 −1 − λ 0 = (A − λ1 I) x = 0 −1 1 2−λ Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 3 −3 −1 x1 0 0 0 0 x2 = 0 −1 1 3 x3 0 We can reduce this to the equivalent system 3 −3 −1 1 1 3 1 1 3 x1 0 0 0 0 = 0 0 0 = 0 0 8 x2 = 0 x3 1 1 3 3 −3 −1 0 0 0 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Solving this system yields the eigenvector 1 x(1) = 1 0 2) For λ2 = 1 2−λ −3 −1 = 1 −3 −1 −1 − λ 0 0 −2 0 = (A − λ2 I) x = 0 −1 1 2−λ −1 1 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 1 −3 −1 1 −3 −1 x1 0 0 −2 0 = 0 −2 0 x2 = 0 0 −2 0 0 0 0 x3 0 Solving this system yields the eigenvector 1 (1) x = 0 1 3) For λ3 = 3 2−λ −3 −1 −1 − λ 0 = (A − λ3 I) x = 0 −1 1 2−λ Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors −1 −3 −1 −1 −3 −1 0 −4 0 = 0 −4 0 = −1 1 1 0 4 0 0 x1 −1 −3 −1 0 −4 0 x2 = 0 0 x3 0 0 0 Solving this system yields the eigenvector −1 x(1) = 0 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Thus, the three linearly independent eigenvectors, are x(1) 1 = 1 0 x(2) 1 = 0 1 Dr. Marco A Roque Sol x(3) − 1 0 = 1 Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Example 7.12 Find the eigenvalues and eigenvectors 4 6 A = 1 3 1 −5 of the matrix 6 2 −2 Solution The eigenvalues λ and eigenvectors x satisfy the equation (A − λI) x = 0, or Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 4−λ 6 6 x1 0 1 3−λ 2 x2 = 0 |A − λI| = −1 −5 −2 − λ x3 0 The eigenvalues are the roots of the equation 4 − λ 4 − λ 6 6 6 6 3−λ 2 = 1 3−λ 2 = |A − λI| = 1 −1 −5 −2 − λ 0 −2 − λ −λ 1 3 − λ 2 − 0 −(4 − λ)(3 − λ) + 6 6 − 2(4 − λ) = −λ3 + 5λ2 − 8λ + 4 = 0 −2 − λ −λ Dr. Marco A Roque Sol −(λ − 1)(λ − 2)2 = 0 Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The roots of are λ1 =, λ2 = 2, and λ3 = 2 . 1) For λ1 = 1 3 6 6 x1 4−λ 6 6 x1 x2 1 2 2 1 3−λ 2 x2 = (A − λ1 I) x = x3 x3 −1 −5 −3 −1 −5 −2 − λ We can reduce this to the equivalent system 3 6 6 x1 1 2 2 x1 1 2 2 x1 0 1 2 2 x2 = 1 2 2 x2 = 1 −3 −1 x2 = 0 0 3 1 x3 0 3 1 x3 0 0 0 x3 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Solving this system yields the eigenvector 4 x(1) = 1 −3 2) For λ2 = 2 4−λ 6 6 2 6 6 x1 1 3−λ 2 1 1 2 x2 (A − λ2,3 I) x = = −1 −5 −2 − λ −1 −5 −4 x3 2 6 6 1 1 2 1 1 2 x1 0 1 1 2 2 x 2 = 0 = 0 4 = 0 4 2 −1 −5 −4Dr. Marco A0Roque −4 0 0Equations 0 x3 0 Sol −2 Ordinary Differential Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The above system is reduced immediately to the equations x1 + x2 + 2x3 = 0 4x2 + 2x3 = 0 Two equations and three unknowns. Hence, one of them 1 is free veriables, let’s say x3 = α, x2 = 21 α, and x3 = −2x3 − x2 = −3α . Thus we have − 3α − 3 1 1 x= = α 2α 2 − 3α − 3 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors In this way, there is one linearly independent eigenvector associated to λ2,3 = 2 , namely, 3 1 x(2) = − 2 Therefore, there are just two linearly independent eigenvectors 1 4 1 x(1) = 1 x(2) = 1 − 3 that is, the matrix A is defective Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Example 7.13 Find the eigenvalues and eigenvectors of the matrix 1 0 0 A = 2 1 −2 3 2 1 Solution The eigenvalues λ and eigenvectors x satisfy the equation (A − λI) x = 0, or Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 1−λ 0 0 x1 0 2 1 − λ −2 x2 = 0 (A − λI) x = 3 2 1−λ x3 0 The eigenvalues are the roots of the equation 1 − λ 0 0 1 − λ −2 = 1 − λ −2 = (1 − λ) |A − λI| = 2 2 1 − λ 3 2 1 − λ (1 − λ)3 + 4(1 − λ) = (1 − λ)(λ2 − 2λ + 5) = 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The roots of are λ1 = 1, λ2 = 1 + 2 i, and λ3 = 1 − 2 i . 1) For λ1 = 1 0 1−λ 0 0 0 0 0 x1 2 1 − λ −2 x2 = 0 (A − λ1 I) x = = 2 0 −2 x3 0 3 2 1−λ 3 2 0 We can reduce this to the equivalent system 2 0 −2 1 0 −1 1 0 −1 x1 0 0 0 0 = 0 0 0 = 0 2 4 x2 = 0 3 2 0 3 2 0 0 0 0 x3 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Solving this system yields the eigenvector 1 x(1) = − 3/2 1 2) For λ2 = 1 + 2 i 1−λ 0 0 x1 1 − λ −2 x2 = (A − λ2 I) x = 2 3 2 1−λ x3 −2 i 0 0 x1 0 2 −2 i −2 x2 = 0 3 2 −2 i x3 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The above system is reduced immediately to the equations x1 = 0 − 2 ix2 − 2x3 = 0, 2x2 − 2 ix3 = 0 Thus, we have one equation and two unknowns. Hence, one of them 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence 0 0 0 0 α 1 =α 1 −i 0 =α x= −i α − i 0 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors In this way, there are two real linearly independent eigenvector associated to λ2 = 1 + 2 i , namely, 0 0 (2) (3) 1 0 x = x = 0 1 3) For λ3 = 1 − 2 i 1−λ 0 0 x1 2 1 − λ −2 x2 = (A − λ2 I) x = 3 2 1−λ x3 2i 0 0 x1 0 2 2 i −2 x2 = 0 3 A2Roque2Soli x3 Differential 0 Equations Dr. Marco Ordinary Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors The above system is reduced immediately to the equations x1 = 0; 2i x2 − 2x3 = 0; 2x2 + 2i x3 = 0 Thus, we have one equation and two unknowns. Hence, one of them 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have 0 0 0 0 1 =α 1 +i 0 x= α =α iα i 0 1 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors In this way, there is two real linearly independent eigenvector associated to λ2 = 1 + 2 i , namely, 0 0 (2) (3) 1 0 x = x = 0 1 Hence, we have three linearly independent eigenvectors, namely 1 0 0 x(1) = − 3/2 x(2) = 1 x(3) = 0 1 0 1 that is, the matrix A is Non-defective Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors OBS Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) are complex eigenvectors of the matrix A with complex eigenvalues λ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for A with eigenvalues λ = u ± i v Finally, let’s introduce another concept from linear algebra The Dot Product For y = (y1 , y2 , ..., yn ), x = (x1 , x2 , ..., xn ) ∈ R, define the dot product or inner product or scalar product.as Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors x · y =< x, y >= x1 x2 y1 y2 . . . xn . = x1 y1 + x2 y2 + ... + xn yn .. yn OBS a) x and y are said to be orthogonal if < x, y >= 0 . b) Orthogonal nonzero vectors ae linearly independent. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Theorem 7.10 Let A be an n × n matrix. If A is symetric, ( A = AT ) then 1) All eigenvalues are real. 2) A is always Nondefective. 3) The eigenvectors corresponding to different eigenvalues are orthogonal, thus if λ1 , λ2 , ..., λn are all simple, v1 , v2 , ..., vn form an orthogonal set. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations The general theory of a system of n first order linear equations x10 = p11 x1 + p12 x2 + . . . + p1n xn + g1 (t) x20 = p21 x1 + p22 x2 + . . . + p2n xn + g2 (t) .. .. . . xn0 = pn1 x1 + pn2 x2 + . . . + pnn xn + gn (t) or X0 = P(t)X + g(t) closely parallels that of a single linear equation of nth order. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations we assume that P and g are continuous on some interval α < t < β; that is, each of the scalar functions p11 , ..., pnn , g1 , ..., gn is continuous there. Theorem 7.4 If the vector functions x(1) and x(2) are solutions of the homogeneus system ( g(t) = 0 ) then the linear combination c1 x(1) + c2 x(2) is also a solution for any constants c1 and c2 . This is the principle of superposition Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations By repeated application of Theorem, we can conclude that if x(1) , ..., x(k) are solutions of the homogeneous system, then c1 x(1) + ... + ck x(k) is also a solution for any constants c1 , ..., ck . Theorem 7.5 If the vector functions x(1) , ..., x(n) are linearly independent solutions of the homogeneous system for each point in the interval α < t < β, then each solution x = phi(t) of the homogeneous system can be expressed as a linear combination of x(1) , ..., x(n) in exactly one way. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations If the constants c1 , ..., cn are thought of as arbitrary, then the above equation includes all solutions of the system, and it is customary to call it the general solution. Any set of solutions x(1) , ..., x(n) of the homogeneus system that is linearly independent at each point in the interval α < t < β is said to be a fundamental set of solutions for that interval. Theorem 7.6 If x(1) , ..., x(n) are solutions of the homogeneus system on the interval α < t < β, then in this interval W [x(1) , ..., x(n) ] either is identically zero or else never vanishes. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations To prove this theorem is necessary to establish that dW = [p11 + p22 + ... + pnn ]W dt Hence W (t) = ce R [p11 +p22 +...+pnn ]dt Theorem 7.7 Let x(1) , ..., x(n) be the solutions of the homogeneus system that satisfy the initial conditions x(1) (t0 ) = e(1) , x(1) (t0 ) = e(2) , ..., x(n) (t0 ) = e(n) , respectively, where t0 is any point inα < t < β and Dr. Marco A Roque Sol Ordinary Differential Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of First Order Linear Equations Basic Theory of Systems of First Order Linear Equations e(1) 1 0 = . .. e(2) 0 1 = . .. 0 ··· e(n) 0 0 = . .. 1 0 Then, x(1) , ..., x(n) form a fundamental set of solutions of the homogeneous system. Finally in the case that the solution is complex-valued, we have the following result. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Basic Theory of Systems of First Order Linear Equations Theorem 7.8 Consider the homogeneous system X0 = P(t)X where each element of P is a real-valued continuous function. If x = u(t) + i v(t) is a complex-valued solution, then its real part u(t) and its imaginary part v(t) are also solutions of this equation. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients We will concentrate most of our attention on systems of homogeneous linear equations with constant coefficients x0 = Ax where A is a constant n × n matrix. Unless stated otherwise, we will assume further that all the elements of A are real (rather than complex) numbers. The case n = 2 is particularly important and lends itself to visualization in the x1x2 − plane, called the phase plane. By evaluating Ax at a large number of points and plotting the resulting vectors, we obtain a direction field of tangent vectors to solutions of the system of differential equations. Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients A qualitative understanding of the behavior of solutions can usually be gained from a direction field. More precise information results from including in the plot some solution curves, or trajectories. A plot that shows a representative sample of trajectories for a given system is called a phase portrait . Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients Now, for the system x0 = Ax we look for solutions of the form x = ve λt where the expon entλ and the vector v are to be determined. Substituting x in the system gives λve λt = Ave λt (A − λI) v = 0 Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients Thus, to solve the system of differential equations, we must solve the above system of algebraic equations. That is, we need to find the eigenvalues and eigenvectors of the matrix A. If we assume that A is a real-valued matrix, then we must consider the following possibilities for the eigenvalues of A: 1. All eigenvalues are real and different from each other. 2. Some eigenvalues occur in complex conjugate pairs. 3. Some eigenvalues, either real or complex, are repeated. If the n eigenvalues are all real and different, as in the Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients Example 7.14 Consider the system 1 1 x = Ax x 4 1 0 Solution Let’s find the eigenvalues of the matrix A Dr. Marco A Roque Sol Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients 1 − λ 1 =0 |A − λI| = 4 1 − λ (1 − λ)2 − 4 = 0 =⇒ (λ2 − 2λ − 3 = (λ − 3)(λ + 1) = 0 =⇒ Dr. Marco A Roque Sol Ordinary Differential Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients λ1 = 3, λ2 = −1 If λ1 = 3, then the system reduces to the single equation −2v1 + v2 = 0, =⇒ v2 = 2v1 and a corresponding eigenvector is v (1) Dr. Marco A Roque Sol 1 = 2 Ordinary Differential Equations Systems of First Order Linear Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Homogeneous Linear Systems with Constant Coefficients Similarly, corresponding to λ2 = −1, we find that a corresponding eigenvector is 1 (2) v = − 2 The corresponding solutions of the differential 1 3t (1) (2) x = e ; x = 2 − Dr. Marco A Roque Sol equation are 1 −t e 2 Ordinary Differential Equations Systems of Linear Algebraic Equations; Linear Independence, Eig Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients The Wronskian of these solutions is 3t e e −t (1) (2) = −4e 2t 6= 0 W [x , x ](t) = 3t 2e −2e −t Hence the solutions x(1) and x(2) form a fundamental set, and the general solution of the system is x= c1 x (1) + c2 x (2) = c1 Dr. Marco A Roque Sol 1 3t e + 2 c2 1 −t e − 2 Ordinary Differential Equations