Chapter2 handout calculus hcmiu sem 1

VIETNAM NATIONAL UNIVERSITY-HCMC International University Chapter Matrices Calculus for Biotechnology Lecturer: Nguyen Minh Quan, PhD quannm@hcmiu.edu.vn Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Contents Linear systems of equations Matrix Operations Inverse matrices Eigenvalues and Eigenvectors Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Introduction Example: Balancing the chemical equations Balancing following chemical equation when the propane gas burns xC3 H8 + yO2 → zCO2 + tH2 O To balance this equation, we can construct a vector as follows: We obtain the linear system of equations         x +y =z +t  2 Q: How to solve this linear system of equations? A: Can use ”matrix” Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Introduction Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier Prof Gilbert Strang, MIT Linear Algebra is extremely useful in a variety of real-world applications, including biology and medicine Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Linear systems of equations We begin with the central problem of linear algebra: solving linear equations Any straight line in the xy-plane can be represented algebraically by an equation of the form a1 x + a2 y = b where a1 , a2 and b are real constants, a1 , a2 not both zero In general, a linear equation in the variables x1 , x2 , , xn is one that can be put in the form a1 x1 + a2 x2 + + an xn = b where a1 , a2 , , an and b are real constants The variables in a linear equation are sometimes called unknowns Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 System of linear equations (n=2) We consider another example 1x + 2y = (1) 4x + 5y = (2) The two unknowns are x and y How to solve? → Elimination Eq (2) − Eq.(1): −3y = −6 → y = Back-substitution x = −1 Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 The Geometry of Linear Equations Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 What is a Matrix? A matrix is a set of elements, organized into rows and columns The number aij , ≤ i ≤ m, ≤ j ≤ n, are called the entries (or elements) of A Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Example of matrix Suppose that a manufacturer has four plants, each of which makes three products If we let aij denote the number of units of product i made by plan j in one week, then the × matrix gives the manufacturer’s production for the week For example, plant makes 270 units of product in one week Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 / 75 Example of matrix The following table gives the nutritional information for some foods It is a × matrix Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 10 / 75 Determinants Determinants of × matrices a11 a12 a13 a21 a22 a23 a31 a32 a33 = a11 a22 a23 a32 a33 − a12 a21 a23 a31 a33 + a13 a21 a22 a31 a32 Example −1 −2 =1 Nguyen Minh Quan (HCMIU-VNU) −1 −2 −5 −1 0 Chapter Matrices +0 −2 = −2 Spring 2015 61 / 75 Determinants of a matrix of order Another method to find the determinant of a 3x3 matrix   a11 a12 a13 A =  a21 a22 a23  a31 a32 a33 det(A) = |A| = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a31 a22 a13 −a32 a23 a11 − a33 a21 a12 Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 62 / 75 Determinants of a matrix of order Example So det(A) = |A| = + 16 − 12 − (−4) − − = Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 63 / 75 Cofactor expansions Let Aij be the matrix formed by removing the ith row and jth column of the matrix A Aij Let Cij := (−1)i+j |Aij |, Cij is called the cofactor of aij Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 64 / 75 The Adjoint and a Theoretical formula for A−1 Definition: Matrix of cofactors The matrix of cofactors of A has the form  C11 C12 · · · C1n  C21 C22 · · · C2n         Cn1 Cn2 · · · Cnn Definition: The Adjoint If A is n × n matrix, the adjoint of A, denoted by adj(A), is the transpose of the matrix of cofactors,   C11 C21 · · · Cn1  C12 C22 · · · Cn2    adj (A) =     C1n C2n · · · Cnn Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 65 / 75 The Adjoint and a Theoretical formula for A−1 The adjoint of A plays the following extremely important role Theoretical formula for A−1 If A is n × n matrix, then Aadj (A) = (det A) I Thus if det A = so that A−1 exists, then A−1 = Nguyen Minh Quan (HCMIU-VNU) adj (A) det A Chapter Matrices Spring 2015 66 / 75 The Adjoint and a Theoretical formula for A−1 Example Let   −3 A =  −2  −1 −3 Thus, det A = −26 and the the matrix of cofactors is   −2 −4 (Cij ) =  −10 −9 −7  12 Therefore,  A−1  −2 −10 1  −9  = adj (A) = det A −26 −4 −7 12 Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 67 / 75 Eigenvalues and Eigenvectors Definition Let A be an n × n matrix An eigenvector of A is a non-zero (column) vector X such that AX = λX , for some scalar (number) λ called eigenvalue Note: If X is an eigenvector associated with λ, then for any number t = 0, the vector tX is also an eigenvector associated with λ Theorem Eigenvalues of A are roots of the equation |A − λI | = and the polynomial p(λ) := |A − λI | is called the characteristics polynomial of A, where I is the n × n identity matrix Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 68 / 75 Eigenvalues and Eigenvectors Example Finding eigenvalues and eigenvectors of the matrix A= 1 −2 Solution: Characteristics polynomial p(λ) = |A − λI | = 1−λ −2 − λ = λ2 − 5λ + λ2 − 5λ + = ⇔ (λ − 2)(λ − 3) = The eigenvalues are λ1 = and λ2 = Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 69 / 75 Find Eigenvectors Find eigenvectors associated with λ1 = Let x1 x2 X = Consider AX = 2X : 1 −2 x1 x2 =2 x1 x2 That is, x1 + x2 = 2x1 −2x1 + 4x2 = 2x2 which reduces to x1 = x2 Choose x2 = t to get, the set of eigenvectors associated with λ1 = has t the form X = =t , for any t = t Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 70 / 75 Find Eigenvectors Find eigenvectors associated with λ1 = Consider AX = 3X : 1 −2 x1 x2 =3 x1 x2 That is, x1 + x2 = 3x1 −2x1 + 4x2 = 3x2 which reduces to 2x1 = x2 Choose x1 = t to get, the set of eigenvectors associated with λ1 = has t the form X = =t , for any t = 2t Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 71 / 75 Exercise Find the eigenvalues and the corresponding eigenvectors of the matrix a A= Hint: λ1 = 5, λ2 = −3 b A= Hint: λ1 = 5, λ2 = −1 3 −6 Hint: λ1 = 3, λ2 = −7 c A= Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 72 / 75 Eigenvalues and Eigenvectors Example Finding eigenvalues and eigenvectors of the matrix   −2  A= −1 −1 Solution: Characteristics polynomial |A − λI| = − λ −2 −1 − λ −1 1 −λ = −(λ − 1)(λ + 1)(λ − 3) = The eigenvalues: λ1 = 1, λ2 = −1, λ3 = Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 73 / 75 Eigenvalues and Eigenvectors Solution (cont.) Find eigenvectors associated with λ1 = 1: Solve (A − λ1 I )X =   −2 1  A − λ1 I =  −1 −1 −1   ∼  −1  0 Solution     −2t −2 X =  t  Thus, eigenvector :t   t Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 74 / 75 Eigenvalues and Eigenvectors Solution (cont.) Find eigenvectors associated with λ2 = −1: Similarly, one can solve (A − λ2 I )X = to get the eigenvector:   t  −1  Find eigenvectors associated with λ3 = 3: Solve (A − λ3 I )X = to get the eigenvector:   −2 t  −1  Nguyen Minh Quan (HCMIU-VNU) Chapter Matrices Spring 2015 75 / 75 ... matrix    1 1 1 1 3R1 +R2 →R2  −6 −3  −−−− −−−−−→  −2R1 +R3 →R3 4 Nguyen Minh Quan (HCMIU- VNU) Chapter Matrices   −3 Spring 2 015 18 / 75 Gaussian elimination  1 1    1 1 −2R2 +R3... are of the  a 11 ± b 11 a12 ± b12  a 21 ± b 21 a22 ± b22  A±B =  same size, then ··· ··· a1n ± b1n a2n ± b2n      am1 ± bm1 am2 ± bm2 · · · amn ± bmn Nguyen Minh Quan (HCMIU- VNU) Chapter... Minh Quan (HCMIU- VNU) Chapter Matrices Spring 2 015 11 / 75 Linear systems of equations Consider the linear system of m equations with n unknowns: a 11 x1 + a12 x2 + + a1n xn = b1 a 21 x1 + a22 x2