5.1 Vectors A vector is a one-dimensional array of numbers. A column vector has its m elements arranged from top to bottom.

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1 Youngstown State University Industrial and Systems Engineering ISEGR Systems Analysis and Design Session - Vector and Matrix Operations. Vectors A vector is a one-dimensional array of numbers. A column vector has its m elements arranged from top to bottom. For example, X X X X A column vector has m rows, each containing one element. The (column) vector X has m rows and one column. One interprets the values according to their positions as X, X, and X. Similarly, the column vector Y Y Y Y Y has three rows and one column. One interprets the values according to their positions as Y, Y, and Y. Also, the vectors R, S 0, and T are all column vectors of length m. The elements in a row vector appear in a horizontal line. In the equations F F F F and G G G G G 9 8 8, F is a row vector of length n, and G is a row vector of length n.. Applications of Vectors A row vector has n columns, each containing one element.

2 Vectors occur naturally in describing physical relationships, business information, manufacturing, process control data,... Indeed, vectors and vector notation provide the most compact way to represent any set of data which describes different instances of a given phenomena. For example, X, X, X may represent the coordinates of a point in space. Similarly, t, t,..., t 0 may represent the times of occurrence of customer orders arriving at a factory. Also, G, G,..., G can store data on the numerical grades earned by students. This course uses vectors in the following contexts: Vectors conveniently represent data addressed by one subscript.. representing data used for statistical process control. adding structure to data by ranking it from smallest to largest. storing values used in computing the area under a function that cannot be integrated analytically.. Vector Addition Vector addition requires the two vectors involved to be of the same size X Y and shape. For example, consider the vectors X X and Y Y. Computing their sum Z requires that corresponding element be added, as expressed by the equation Z j X j + Y j, j,,. For the values of X and Y given above, Z X + Y X 8 Y Vector addition requires that corresponding elements be added. As a result, two vectors must be the same shape so that no elements are left over in their sum... Vector Transposition One obtains the transpose X t of a column vector X having m elements by writing its column as a row. For example, X X t and Y leads to Xt To transpose a column vector, write it as a row. To transpose a row vector, write it as a column.

3 .. Vector Multiplication The definition of the product of two vectors requires that the number of columns in the first equal the number of rows in the second. One computes the product of a row vector U of length n and a column vector V of length n via U V U U... U n In the examples that follow, V V... V n A, B, X The product W A X requires the computations W A X The product Z B Y requires the computations Z B Y n U j V j U V +U V +... U n V n, or,. j, and Y. A X +A X +A X + +. B Y +B Y +B Y + +. Vector multiplication also imposes requirements on the shape of the vectors being multiplied.. Matrices A matrix C is a two-dimensional array of numbers. Convention requires the use of m to denote the number of rows and n to denote the number of columns. For example, a matrix A having two rows and three columns (m and n ) is Matrices conveniently A A A A A A A 8 Matrix multiplication also requires that the number of columns in the first equal the number of rows in the second. More on matrices later. represent data addressed by two subscripts.

4 Read the matrices by corresponding positions to observe that (for example) A, A, and A. As a second example, consider the following by matrix B. B B B B B B B 9 Read the matrices by corresponding positions to observe that (for example) B, B, and B. The next example uses a by matrix D: D D D D D D D D D D D D D 9 9 Read the matrices by corresponding positions to observe that D 9, D, and D... Applications of Matrices Matrices occur naturally in describing phenomena indexed by two parameters. For example, E ij conveniently represents the elevation of a point having longitude i and latitude j. As a second example, using matrix notation to represent the coefficients of the variables on the left side of the system of linear equations X + 8X + X X X X X + X + 8X results in both conceptual and mathematical advantages. Similarly, a matrix conveniently represents data for a bill of materials. For example, consider the fabrication of assemblies j from parts, and. A parts requirements matrix links the number of assemblies to the number of parts as follows: P P P P P P P P P P 0 0 The matrix elements P ij give the number of units of part i that are required to make assembly j. One uses the number of Type j assemblies with the P matrix to determine the number of parts required. For this example, making

5 two Type assemblies requires P of Part, P 0 0 of Part, and P of Part. This course uses matrices in representing (i) statistical process control data, (ii) solving simultaneous equations, and (iii) representing bill of materials data... Matrix Addition One computes the elements for the sum C of two m by n matrices A and B via the equations C ij A ij + B ij, i,,... m and j,,... n. Note that this one equation is used for all possible pairs of combinations of values of i and j, where i ranges over the integers in the set {,,..., m} and j ranges over the integers in the set {,,..., n} One uses the equation m n times. Example Sum of Matrices 8 A Given: Their sum C A + B is computed as B 9 Matrix addition requires that the two matrices must have the same number of rows and the same number of columns. This assures that no elements are left over when the addition is performed. Note that addition is commutative. A + B B + A. C A + B Matrix Transposition The transpose X t of a m by n matrix X results from interchanging the rows and columns of X. The equation X t ji X ij, i,,..., m and j,,... n. expresses the relationship in formal mathematical terms. For example, the transpose of is 0 Also, G, G,..., G can store data on the numerical grades earned by students. This course uses vectors in the following contexts:.

6 representing data used for statistical process control adding structure to data by ranking it from smallest to largest storing values used in computing the area under a function that cannot be integrated analytically.. Matrix Multiplication Let A denote a matrix having m A rows and n A columns, and B denote a m B by n B matrix. Multiplication A B is defined for n A m B. The equation C ij n A k A ik B kj, i,,... m A and j,,..., n B expresses the computations required to compute the product of two (correctly shaped) matrices. For example, A and B have the product C A B As another example, the product of F is given by H F G and G. 0 Areas of Application Matrix multiplication applies in a rich variety of scenarios having data addressed by two indices. For example, consider a bill of materials for a manufactured assembly. Assembly requires of Part, of Part, of Part and of Part. The matrix P defined next displays similar requirements for all three assemblies. P 0 0.

7 Element P ij gives the number of Type i parts that go into the fabrication of one unit of Assembly j. If the demand for the number of assemblies that D must be produced is given by D D then the total number D of Type i parts needed to meet the demand is 0 0 Linear Equations One can solve the set of linear equations X + 8X + X X X X X + X + 8X by using vector and matrix operations. A X B, where 8 X A, X X 8 X, and B Express the set of equations by. One expresses the solution to the system of linear equations by X A B, where A denotes the inverse of A. Calculation of the inverse of a square matrix is considered next.. Exercises.. Exercise.. Making a unit of product requires of part R, of part S, and of part T. Making a unit of product requires of part R, of part S, and of part T. Set up Excel to allow for a vector of product demands to be translated into parts requirements. Making P j, j, units requires ( P of each type of part. For example, making 0 of Product and of Product requires P )

8 Figure : Excel Setup ( 0 ) For example, making of Product and 0 of Product requires ( ) Enter the matrix of technology constraints into Excel. To the right of that matrix, enter a column vector giving the demands for products. Compute the parts required as specified in the vectors of product requirements given below. ( P P ( ) 00 and. 00 ) ( 0 ) ( 0. Visual Basic The ) ( ) ( ) ( 0 0 Sub MatrixMult() Dim A(, ) As Single, P() As Single, Req() As Single Dim prow As Integer, pcol As Integer, Sum As Single, Term As Single Dim I As Integer, J As Integer, K As Integer, M As Integer, N As Integer adsf 8 ),

9 Figure : Excel - Block Equation Entry Acquire A matrix For I To prow I + For J To pcol J + A(I, J) Cells(pRow, pcol) Next J Next I display A on sheet For I To prow I + For J To pcol J + Sheets().Cells(pRow, pcol) A(I, J) Next J Next I Sheets().Activate 9

10 Acquire P matrix pcol 8 For I To prow I + P(I) Cells(pRow, pcol) Next I compute product For I To Sum 0 For K To Term A(I, K) * P(K) Sum Sum + Term Next K Req(I) Sum Next I display Req on sheet For I To prow I + pcol J + Sheets().Cells(pRow, pcol) Req(I) Next I Sheets().Activate End Sub 0