Matrix Assignment

Application of Matrix in Engineering

Concept of Matrix or Matrices

Let us now introduce the concept of a matrix. Consider a set of scalar quantities arranged in a

rectangular array containing m rows and n columns:

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This array will be called a rectangular matrix of order m by n, or, briefly, an m × n matrix. Not

every rectangular array is a matrix; to qualify as such it must obey the operational rules discussed

below.

The quantities ai j are called the entries or components of the matrix. Preference will be given to

the latter unless one is talking about the computer implementation. As in the case of vectors, the

term “matrix element” will be avoided to lessen the chance of confusion with finite elements. The

two subscripts identify the row and column, respectively.

Matrices are conventionally identified by bold uppercase letters such as A, B, etc. The entries of

matrix A may be denoted as Ai j or ai j , according to the intended use. Occassionally we shall use

the short-hand component notation

A = [ai j ].

Where Do Matrices Come From?

Although we speak of “matrix algebra” as embodying vectors as special cases of matrices, in practice the quantities of primary interest to the structural engineer are vectors rather than matrices. For example, an engineer may be interested in displacement vectors, force vectors, vibration eigenvectors, buckling eigenvectors. In finite element analysis even stresses and strains are often arrangedas vectors although they are really tensors.

On the other hand, matrices are rarely the quantities of primary interest: they work silently in the

background where they are normally engaged in operating on vectors.

Why use Matrices?

We use matrices in mathematics and engineering because often we need to deal with several

variables at once—e.g. the coordinates of a point in the plane are written (x, y) or in space as

(x, y, z) and these are often written as column matrices in the...