Matrices and matrix operations

Matrices

In the lesson of the course Algebra and Geometry I on first-degree equations and linear systems, Gaussian elimination was carried out on an array that gathered the coefficients of each equation: the unknowns did not intervene in the computation, and all the work was done with the numbers of the array. That observation is old; The Nine Chapters on the Mathematical Art, a Chinese treatise compiled about two thousand years ago, already solved systems by arranging the coefficients in columns. The word matrix was introduced by James Joseph Sylvester in 1850, and Arthur Cayley published in 1858 the first theory of operations with matrices, which is the one this lesson studies.

Let $m$ and $n$ be natural numbers. A matrix of size $m \times n$ is an array of $mn$ numbers arranged in $m$ rows and $n$ columns. The number located in row $i$ and column $j$ is called the entry $(i, j)$ of the matrix; if the matrix is denoted by a capital letter, such as $A$, the entry is denoted by the corresponding lowercase letter with two subscripts, $a_{ij}$, the first of which always indicates the row. One also writes $A = (a_{ij})$. A $2 \times 3$ matrix has, for example, the following form:

The entries will be real numbers unless otherwise stated; everything that follows holds without change for matrices of complex numbers, because the proofs only use the properties of the addition and the multiplication of numbers, which $\mathbb{C}$ shares with $\mathbb{R}$. Two matrices are equal when they have the same size and the same entries in the same positions: an equality between $m \times n$ matrices amounts to $mn$ equalities between numbers.

Types of matrices

Some shapes occur so often that they have names of their own.

• A row matrix is a $1 \times n$ matrix, and a column matrix is a $m \times 1$ matrix. The vectors in the plane of the Course I lesson on vectors in the plane can be written as $2 \times 1$ column matrices.
• The zero matrix of a given size has all its entries equal to $0$; it is also denoted by $0$, and the context indicates its size.
• A matrix is square if it has as many rows as columns; an $n \times n$ matrix is also called a matrix of order $n$. Its entries $a_{11}, a_{22}, \ldots, a_{nn}$ form the main diagonal.
• A square matrix is diagonal if all its entries off the main diagonal are zero. The identity matrix of order $n$, which is written $I_n$, or $I$ when the order is clear, is the diagonal matrix whose diagonal entries equal $1$.
• A square matrix is upper triangular if all the entries located below the main diagonal are zero, that is, if $a_{ij} = 0$ whenever $i > j$; it is lower triangular if those located above it are zero.
• A square matrix is symmetric if $a_{ij} = a_{ji}$ for all indices: the main diagonal is an axis of symmetry of the array.

For example, of the four matrices below, the first is the identity of order $3$, the second is diagonal, the third is upper triangular and the fourth is symmetric.