Presentation on theme: "CHAPTER ONE Matrices and System Equations"— Presentation transcript:
1 CHAPTER ONE Matrices and System Equations Objective:To provide solvability conditions of a linear equation Ax=b and introduce the Gaussian elimination method, a systematical approach in solving Ax=b, to solve it.
2 Outline Motivative Example. Elementary row operations and Elementary Matrices.Some Basic Properties of Matrices.Gaussian Elimination for solving Ax=b.Solvability conditions for Ax=b.
3 Motivative Example (curve fitting) Given three points( )( )( ),find a polynomial of degree 2 passing through the three given points.Solution: Let the polynomial beWhere a,b and c are to be determinedAx=b
4 Question: Why transform to matrix form? To provide a systematic approach and to use computer resource.
5 Question: How to solve Ax=b systematically? One way is to put Ax=b in triangular form,whichcan be easily solved by back-substitution.Definition: A system is said to be in triangular form if in the k-th equation the coefficients of thee first (k-1) variables are all zero and the coefficient of xk is nonzero ( k = 1,…,n)
7 Solution: By elementary row operations as described below. Question: How to put Ax=b in triangular form while leaving the solution set invariant?Solution: By elementary row operations as describedbelow.Definition: Two systems of equations involing the same variables are said to be equivalent if they have the same solution set.
8 Before introducing elementary operation, we recall some definitions and notations. (§ 1.3)Equality of two matrices.Multiplication of a matrix by a scalar.Matrix addition.Matrix multiplication.Identity matrix.Multiplicative inverse.Nonsingular and singular matrix.Transpose of a matrix.
9 Definitions Def. If and , then the where . Matrix Multiplication ,whereDef. An (n × n) matrix A is said to be nonsingularor invertible if there exists a matrix B such thatAB=BA=I. The matrix B is said to be amultiplicative inverse of A. And B is denotedby A-1.Warning: In general, AB≠BA. Matrix multiplication is not commutative.
10 Definitions (cont.)Def. The transpose of an (m × n) matrix A is the (n ×m) matrix B defined by for j=1,…,n andi=1,…,m. The transpose of A is denoted by AT.Def. An (n × n) matrix A is said to be symmetric if AT=A .
11 Some Matrix Properties Let be scalars,A,B and C be matriceswith proper dimensions.(Commutative Law)(Associative Law)(Distributive Law)
15 Def: Let and Thenis said to be a linear combination ofNote that We have the next result.Theorem1.3.1: Ax=b is consistent b can be writtenas a linear combination of colum vectorsof A.
16 Calories Burned Per Hour Application 1: Weight ReductionTable 1Calories Burned Per HourWeight in lbExercise Activity152161170178Walking 2 mph213225237249Running 5.5 mph651688726764Bicycling 5.5mph304321338356Tennis420441468492
17 Hours Per Day For Each Activity Application 1: Weight Reduction (cont.)Table 2Hours Per Day For Each ActivityExercise schedulewalkingRunningBicyclingTennisMonday1.00.0Tuesday2.0Wednesday0.40.5ThursdayFriday
23 Amount Produced Per Quarter Application 2: Production Costs (end)Solution:Table 5Amount Produced Per QuarterSeasonSummerFallWinterSpringYearRaw materials1,8702,1602,0701,9608,060Labor3,4503,9403,8103,58014,780Overhead and miscellaneous1,6701,9001,8301,7407,140Total production cost6,9908,0007,7107,28029,980
25 Application 5: Networks and Graphs (cont.) DEF.
26 Application 5: Networks and Graphs (end) TheoremIf A is an n × n adjacency matrix of a graph and representsthe ijth entry of Ak, then is equal to the number of walks of lengthfrom to Vi to Vj.
27 Application 6: Information Retrieval (P.59) Suppose that our database, consists of these book titles:B1. Applied Linear AlgebraB2. Elementary Linear AlgebraB3. Elementary Linear Algebra with ApplicationsB4. Linear Algebra and Its ApplicationsB5. Linear Algebra with ApplicationsB6. Matrix Algebra with ApplicationsB7. Matrix TheoryThe collection of key words is given by the following alphabeticallist：algebra, application, elementary, linear, matrix, theory
28 Array Representation for Database of Linear Algebra Books Application 6: Information Retrieval (cont.)Table 8Array Representation forDatabase of Linear Algebra BooksBooksKey WordsB1B2B3B4B5B6B7algebra1applicationelementarylinearmatrixtheory
29 Application 6: Information Retrieval (end) If the words we are searching for are applied, linear, and algebra,then the database matrix and search vector are given byIf we set y= ATx, then
31 Three types of Elementary row operations. (§ 1.2)Three types of Elementary row operations.I. Interchange two row.II. Multiply a row byIII. Replace a row by its sum with a multiple ofanother row.
32 Lead variables and free variables(p.15) Eg:, and are lead variables while andare free variables.
33 Def. A matrix is said to be in row echelon form if (i) The first nonzero entry in each row is 1.(ii) If row k does not consist entirely of zero,the number of leading zero entries in rowk+1 is grater then the number of leadingzero entries in row k.(iii) If there are rows whose entries are all zero, theyare below the rows having nonzero entries.Def. The process of using row operations I, II, and III totransform a linear system into one whose augmentedmatrix is in row echelon form is called Gaussianelimination.
34 Overdetermined and Underdetermined Def. A linear system is said to be overdeterminedif there are more equations(m) than unknowns(n). (m > n)Warning: Overdetermined systems are usually (but not always) in consistent.Def. A system of m linear equations in n unknownsis said to be underdetermined if there arefewer equations. (m < n)
35 Reduced Row Echelon Form Def. A matrix is said to be in reduced row echelonform if:(i) The matrix is in row echelon form.(ii) The first nonzero entry in each row is theonly nonzero entry in its column.Def. The process of using elementary row operationsto transform a matrix into reduced row echelonform is called Gauss-Jordan reduction.
37 Application 2: Electrical Networks (end) Kirchhoff’s Laws:1. At every node the sum of the incoming currents equals thesum of the outgoing currents.2. Around every closed loop the algebraic sum of the voltagemust equal the algebraic sum of the voltage drops.
38 Application 4: Economic Models For Exchange of Goods (P.25) F M CFMC1/21/41/31/21/4
39 (§ 1.4) Elementary Matrices Type I ( ): Obtained by interchanging rows i and jfrom identity matrix.Type II ( ): Obtained from identity matrix bymultiplying row i withType III ( ): Obtained from identity matrix by addingto row j.
40 Elementary Row / Column Operation means performing type I row operation on A.means performing type II row operation on A.means performing type III row operation on A.means performing type I column operation on A.means performing type II column operation on A.means performing type III column operation on A.
41 Theorem1.4.2:If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.WithThe solution set of a linear equations is invariant under three types row operation. and have the solution set.
42 Def. A matrix B is row equivalent to A if there exists a Row Equivalent (P.71)Def. A matrix B is row equivalent to A if there exists afinite sequence of elementary matrices such thatTheorem1.4.3(a) A is nonsingular.(b) Ax=0 has only the trivial solution 0.(c) A is row equivalent to I.
43 Proof of Theorem 1.4.3 (a) (b) Let be a solution of Ax=0. (b) (c) Let A ~ U, where U is in reduced row echelon form.Suppose U contains a zero row.by Th1.2.1, Ux=0 has a nontrivial solutionthus A~I.(c) (a)A~I A= E1 …… Ek for some E1 … Ek∵ each Ei is nonsingular.∴ A is nonsingular. (by Th.1.2.1)row
44 Corollary1.4.4 Ax=b has a unique solution A is nonsingular. Pf: " “ The unique solution is" " Suppose is the unique solution and A issingular.is also a solution of Ax=b.A is nonsingular.
45 BUT in general, and AB=AC B=C. Eg.Moreover,AC=AB while
46 Method For Computing = (I | Ek…E1‧I) ( by ) = (I | A-1) If A is nonsingular and row equivalent to I, sothere exists elementary matrices such thatthen,Ek…E1(A | I)= (Ek…E1‧A | Ek…E1‧I) ( by )= (I | Ek…E1‧I) ( by )= (I | A-1)2004 NCTU ECE Linear Algebra
47 Example 4. (P.73) Sol: Q: Compute A-1 if . 2004 NCTU ECE Linear Algebra
48 Example 4. (cont.) Sol: Q: Compute A-1 if . 2004 NCTU ECE Linear Algebra
49 Diagonal and Triangular Matrices Def. An n × n matrix A is said to be upper triangular if aij=0 for i > j and lower triangular if aij=0 for i > j.Def. An n × n matrix B is diagonal if aij=0 whenever i ≠ j.Triangular FactorizationIf an n × n matrix C can be reduced to upper triangular formusing only row operation III, then C has an LU factorization.The matrix L is unit lower triangular, and if i > j, then lij is the multiple of t he jth row subtracted from the ith row during the reduction process.2004 NCTU ECE Linear Algebra
50 Example 6. (P.74)row operation IIIMark:2004 NCTU ECE Linear Algebra
51 Block MultiplicationLet A be an m × n matrix and B is an n × r matrix.It is often useful to partition A and B and express theproduct in terms of the submatrices of A and B.In general, partition B into columnsthenpartition A into rows , then2004 NCTU ECE Linear Algebra
52 Block Multiplication (cont.) Case 1.Case 2.Case 3.2004 NCTU ECE Linear Algebra
53 Block Multiplication (cont.) Case 4.Letthen2004 NCTU ECE Linear Algebra
54 Example 2. (P.85) Let A be an n × n matrix of the form , where A11 is a k × k matrix (k < n ) .Show that A is nonsingular if and only if A11 and A22are nonsingular.Solution:2004 NCTU ECE Linear Algebra
55 Scalar / Inner Product Give two vectors , This product is referred to as a scalar product or aninner product.2004 NCTU ECE Linear Algebra
56 Outer Product Give two vectors , The product is referred to as the outer productof2004 NCTU ECE Linear Algebra
57 Outer Product Expansion Suppose that , thenThis representation is referred to as an outer productexpansion .2004 NCTU ECE Linear Algebra