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MATRICES

Matrices are an efficient way to represent a set of data using a series of rows and columns. The individual values are scalars, the individual rows can be considered as vectors, and the columns as components of those vectors. Consider the matrix,

A = ⎡⎢⎣ 1 2 3       4 5 6 ⎤⎥⎦

The dimensions of a matrix are given in terms of “m” and “n”, with “m” signifying the number of rows and “n” the number of columns. Matrices are always defined by their rows and then their columns, so matrix “A” would be a “2 × 3” matrix. Each element in a matrix can be defined by its column and row location. For example, the elements in matrix A can be defined as Amn, signifying the element in row M and column n.


SQUARE MATRICES
A square matrix is one where rows and columns equal each other (m=n). For example, a 3x3 matrix is

⎡⎢⎢⎢⎣ 1 1 1       1 1 1       1 1 1 ⎤⎥⎥⎥⎦


SYMMETRIC MATRICES
If the values of the elements in the matrix are the same if the row and column identifiers are switched, then the matrix is a symmetric matrix. Mathematically, this means that Amn = Anm. An example of a symmetric matrix is

⎡⎢⎢⎢⎣ 1 2 3       2 1 4       3 4 1 ⎤⎥⎥⎥⎦


DIAGONAL MATRICES
A diagonal matrix has values only when m = n. This results in non-zero elements along the diagonal of the matrix:

⎡⎢⎢⎢⎣ 1 0 0       0 1 0       0 0 1 ⎤⎥⎥⎥⎦

In this instance, most matrix operations are simplified into simple algebraic operations.


UPPER TRIANGULAR MATRICES
An upper triangular matrix is a matrix where all of the values below the “diagonal” are zeros:

⎡⎢⎢⎢⎣ 1 1 1       0 1 1       0 0 1 ⎤⎥⎥⎥⎦


LOWER TRIANGULAR MATRICES
A lower triangular matrix is a matrix where all the values above the “diagonal” are zeros:

⎡⎢⎢⎢⎣ 1 0 0       1 1 0       1 1 1 ⎤⎥⎥⎥⎦


VECTORS AS MATRICES
Often a single vector will be expressed as either a column or row matrix and groups of “like” vectors will be grouped together as a single matrix. For example vectors A and B

(1) A = 5 i + 6 j + 7 k   B =  − 6 i − 12 k

can be represented as

⎡⎢⎣    5 6   7        − 6   0  − 7 ⎤⎥⎦

where each column represents a component (i, j, k) and each row as separate vector. Note that the lack of a j component in the second vector necessitates a “0” for the corresponding place in the matrix.
It is important to understand that by converting vectors into matrices then mathematical operations and solving a system of equations becomes more efficient.