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Linear Programming and Resource Allocation Modeling
book

Linear Programming and Resource Allocation Modeling

by Michael J. Panik
November 2018
Intermediate to advanced content levelIntermediate to advanced
448 pages
12h 24m
English
Wiley
Content preview from Linear Programming and Resource Allocation Modeling

2Mathematical Foundations

2.1 Matrix Algebra

We may define a matrix as an ordered set of elements arranged in a rectangular array of rows and columns. Thus a matrix A may be represented as

images

where aij, i = 1, …, m; j = 1, …, n, is the (representative) element in the ith row and jth column of A. Since there are m rows and n columns in A, the matrix is said to be “of order m by n” (denoted (m × n)). When m = n, the matrix is square and will simply be referred to as being “an nth order” matrix. To economize on notation, we may represent A in the alternative fashion

images

Oftentimes we shall need to utilize the notion of a matrix within a matrix, i.e. a submatrix is the (k × s) matrix B obtained by deleting all but k rows and s columns of an (m × n) matrix A.

Let us now examine some fundamental matrix operations. Specifically, the sum of two (m × n) matrices A = [aij], B = [bij] is the (m × n) matrix A + B = C = [cij], where cij = aij + bij, i = 1, …, m; j = 1, …, n, i.e, we add corresponding elements. Next, to multiply an (m × n) matrix A by a scalar λ we simply multiply each element of the matrix by the scalar or

images

(In view of these operations it is evident that A − B = A + (−1) B =  ...

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Publisher Resources

ISBN: 9781119509448Purchase book