Presentation on theme: "Matrices and Systems of Equations"— Presentation transcript:
1 Matrices and Systems of Equations Chapter 1Matrices and Systems of Equations
2 Systems of Linear Equations Where the aij’s and bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.
3 Definition Example Inconsistent : A linear system has no solution. Consistent : A linear system has at least one solution.Example(ⅰ) x1 + x2 = 2x1 − x2 = 2(ⅱ) x1 + x2 = 2x1 + x2 =1(ⅲ) x1 + x2 = 2−x1 − x2 =-2
4 Three Operations that can be used on a system DefinitionTwo systems of equations involving the same variables are said to be equivalent if they have the same solution set.Three Operations that can be used on a systemto obtain an equivalent system:Ⅰ. The order in which any two equations are written maybe interchanged.Ⅱ. Both sides of an equation may be multiplied by thesame nonzero real number.Ⅲ. A multiple of one equation may be added to (or subtracted from) another.
5 is in strict triangular form. n×n SystemsDefinitionA system is said to be in strict triangular form if in the kthequation the coefficients of the first k-1 variables are allzero and the coefficient of xk is nonzero (k=1, …,n).Example The systemis in strict triangular form.
9 Definition A matrix is said to be in row echelon form ⅰ. If the first nonzero entry in each nonzero row is 1.ⅱ. If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k.ⅲ. If there are rows whose entries are all zero, they are below the rows having nonzero entries.
10 Example Determine whether the following matrices are in row echelon form or not.
11 Definition Definition The process of using operations Ⅰ, Ⅱ, Ⅲ to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.DefinitionA linear system is said to be overdetermined if there are more equations than unknows.A system of m linear equations in n unknows is said to be underdetermined if there are fewer equations than unknows (m<n).
13 Definition A matrix is said to be in reduced row echelon form if: ⅰ. The matrix is in row echelon form.ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.
14 Homogeneous SystemsA system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero.Theorem An m×n homogeneous system of linear equations has a nontrivial solution if n>m.
17 Scalar Multiplication DefinitionTwo m×n matrices A and B are said to be equal if aij=bij for each i and j.Scalar MultiplicationIf A is a matrix and k is a scalar, then kA is the matrixformed by multiplying each of the entries of A by k.DefinitionIf A is an m×n matrix and k is a scalar, then kA is the m×n matrix whose (i, j) entry is kaij.
18 Matrix Addition Definition Two matrices with the same dimensions can be addedby adding their corresponding entries.DefinitionIf A=(aij) and B=(bij) are both m×n matrices,then the sum A+B is the m×n matrix whose (i, j) entry is aij+bij for each ordered pair (i, j).
20 cij = ai1b1j + ai2b2j +…+ ainbnj = aikbkj. Matrix MultiplicationDefinitionIf A=(aij) is an m×n matrix and B=(bij) is an n×r matrix, then the product AB=C=(cij) is the m×r matrix whose entries are defined bycij = ai1b1j + ai2b2j +…+ ainbnj = aikbkj.k=1n
21 Examplethen calculate AB.1. Ifthen calculate AB and BA.2. If
22 Matrix Multiplication and Linear Systems Case 1 One equation in Several UnknowsIf we let andthen we define the product AX by
23 Case 2 M equations in N Unknows If we let andthen we define the product AX by
24 DefinitionIf a1, a2, … , an are vectors in Rm and c1, c2, … , cn are scalars, then a sum of the formc1a1+c2a2+‥‥cnanis said to be a linear combination of the vectors a1, a2, … , an .Theorem (Consistency Theorem for Linear Systems)A linear system AX=b is consistent if and only if b can be written as a linear combination of the column vectors of A.
25 Theorem Each of the following statements is valid for any scalars k and l and for any matrices A, B and C for which the indicated operations are defined.A+B=B+A(A+B)+C=A+(B+C)(AB)C=A(BC)A(B+C)=AB+AC(A+B)+C=AC+BC(kl)A=k(lA)k(AB)=(kA)B=A(kB)(k+l)A=kA+lAk(A+B)=kA+kB
26 The Identity MatrixDefinitionThe n×n identity is the matrix where
27 Matrix Inversion Definition Definition An n×n matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I.Then matrix B is said to be a multiplicative inverse of A.DefinitionAn n×n matrix is said to be singular if it does not have a multiplicative inverse.
28 The Transpose of a Matrix Theorem If A and B are nonsingular n×n matrices, then AB is also nonsingular and (AB)-1=B-1A-1The Transpose of a MatrixDefinitionThe transpose of an m×n matrix A is the n×m matrix B defined bybji=aijfor j=1, …, n and i=1, …, m. The transpose of A is denoted by AT.
29 Algebra Rules for Transpose: (AT)T=A(kA)T=kAT(A+B)T=AT+BT(AB)T=BTATDefinitionAn n×n matrix A is said to be symmetric if AT=A.
30 4. Elementary MatricesIf we start with the identity matrix I and then perform exactly one elementary row operation, the resulting matrix is called an elementary matrix.
31 Type I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I.Example Letand let A be a 3×3 matrixthen
32 Type II. An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant.Example Letand let A be a 3×3 matrixthen
33 Type III. An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.Example Letand let A be a 3×3 matrix
34 In general, suppose that E is an n×n elementary matrix In general, suppose that E is an n×n elementary matrix. E is obtained by either a row operation or a column operation.If A is an n×r matrix, premultiplying A by E has theeffect of performing that same row operation on A. If Bis an m×n matrix, postmultiplying B by E is equivalentto performing that same column operation on B.
35 ExampleLet,Find the elementary matrices ， ，such that
36 Theorem If E is an elementary matrix, then E is nonsingular and E-1 is an elementary matrix of the same type.DefinitionA matrix B is row equivalent to A if there exists a finite sequence E1, E2, … , Ek of elementary matrices such thatB=EkEk-1‥‥E1A
37 Theorem 1.4.2 (Equivalent Conditions for Nonsingularity) Let A be an n×n matrix. The following are equivalent:A is nonsingular.Ax=0 has only the trivial solution 0.A is row equivalent to I.Theorem The system of n linear equations in n unknowns Ax=b has a unique solution if and only if A is nonsingular.
38 A method for finding the inverse of a matrix If A is nonsingular, then A is row equivalent to I andhence there exist elementary matrices E1, …, Ek suchthatEkEk-1‥‥E1A=Imultiplying both sides of thisequation on the right by A-1EkEk-1‥‥E1I=A-1row operationsThus (A I)(I A-1)
41 Diagonal and Triangular Matrices An n×n matrix A is said to be upper triangular if aij=0 for i>jand lower triangular if aij=0 for i<j.A is said to be triangular if it is either upper triangular orlower triangular.An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .
43 In general, if A is an m×n matrix and B is an n×r that has been partitioned into columns (b1, …, br), then the blockmultiplication of A times B is given byAB=(Ab1, Ab2, … , Abr)If we partition A into rows, thenThen the product AB can be partitioned into rows as follows:
44 Case 1 B=(B1 B2), where B1 is an n×t matrix and B2 Block MultiplicationLet A be an m×n matrix and B an n×r matrix.Case 1 B=(B1 B2), where B1 is an n×t matrix and B2is an n×(r-t) matrix.AB= A(b1, … , bt, bt+1, … , br)= (Ab1, … , Abt, Abt+1, … , Abr)= (A(b1, … , bt), A(bt+1, … , br))= (AB1 AB2)
45 Case 2 A= ,where A1 is a k×n matrix and A2 is an (m-k)×n matrix.ThusCase 3 A=(A1 A2) and B= , where A1 is an m×s matrixand A2 is an m×(n-s) matrix, B1 is an s×r matrix and B2 is an(n-s)×r matrix.Thus
46 Case 4 Let A and B both be partitioned as follows： B11 B sB=B21 B n-st r-tA11 A kA=A21 A m-ks n-sThen
47 In general, if the blocks have the proper dimensions, the block multiplication can be carried out in the same manner as ordinary matrix multiplication.