Matrix Algebra

Matrices: Definition

A matrix is a set of numbers arranged in a table. For example, Toto, Marius, and Olivette are looking at their possessions, and they are counting how many balls, cars, coins, and novels they each possess. Toto has 2 balls, 5 cars, 10 coins, and 20 novels. Marius has 1, 2, 3, and 4, and Olivette has 6, 1, 3, and 10. These data can be displayed in a table in which each row represents a person and each column a possession:

We can also say that these data are described by the matrix denoted A equal to

Matrices are denoted by boldface uppercase letters.

To identify a specific element of a matrix, we use its row and column numbers. For example, the cell defined by row 3 and column 1 contains the value 6. We write that a3,1 = 6. With this notation, elements of a matrix are denoted with the same letter as the matrix but written in lowercase italic. The first subscript always gives the row number of the element (i.e., 3), and second subscript always gives its column number (i.e., 1).

A generic element of a matrix is identified with indices such as i and j. So, aij is the element at the ith row and jth column of A. The total number of rows and columns is denoted with the same letters as the indices but in uppercase letters. The matrix A has I rows (here I = 3) and j columns (here j = 4), and it is made of I × j elements aij (here 3×4 = 12). The term dimensions is often used to refer to the number of rows and columns, so A has dimensions I by j.

As a shortcut, a matrix can be represented by its generic element written in brackets. So A with I rows and j columns is denoted

For either convenience or clarity, the number of rows and columns can also be indicated as subscripts below the matrix name:

Vectors

A matrix with one column is called a column vector or simply a vector. Vectors are denoted with bold lowercase letters. For example, the first column of matrix A (of Equation 1) is a column vector that stores the number of balls of Toto, Marius, and Olivette. We can call it b (for balls), and so

Vectors are the building blocks of matrices. For example, A (of Equation 1) is made of four column vectors, which represent the number of balls, cars, coins, and novels, respectively.