Matrix Algebra - Fundamentals

The following are a few basic definitions concerning matrices.

Definition. A matrix is a rectangularly shaped array with m rows and n columns of mn mathematical objects of a given basic set.

The order of a matrix is mn ("m by n"). Each column and each row of a matrix defines a vector. A column vector is nothing other than an m1 matrix, and a row vector is a 1n matrix.

Matrices are denoted by bold uppercase letters, e.g.  A. Matrix elements are denoted by lowercase letters subscripted by two indices, i.e. a_m,n. Sometimes the comma between the indices is omitted. The sequence of the indices is not arbitrary; the first index always denotes the row, the second index the column. If m=n, the matrix is called a square matrix of order n. If a matrix is square, the diagonal containing elements of equal indices (a₁₁, a₂₂, ..., a_nn) is called the principal diagonal of this matrix. The trace of a matrix is the sum of all elements of the principal diagonal.