Matrices

An m by n matrix over a set S is a rectangular array of elements of S, arranged in m rows and n columns. It is customary to write an m by n matrix using notation such as

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ \dots & \dots & \dots & \dots \\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right)$$

The m x n matrix A can also be written compactly as A [a_ij]_mxn or simply as A = [a_ij] or A = (a_ij), if the dimension is known from the context.

Fixed the i-esima row (or j-esima column) of a matrix A = (a_ij), the elements a_i1,a_i2, a_in can be seen as vectors of ℝⁿ. For this reason thye are known as row (column) vectors and are indicated as:

$${\mathbf{a}}_{\mathbf{i}}=(a_{i1},a_{i2},\dots ,a_{in})or{\mathbf{a}}^{\mathbf{j}}=\left(\begin{array}{c}{a}_{i1}\\ a_{i2}\\ \dots \\ a_{mj}\end{array}\right)$$

Some Special Matrices

A matrix A = [a_ij]_n×n, which has an equal number of rows and columns, is called a square matrix of order n. Otherwise it is called rectangular.

The elements a₁₁, a₂₂, ..., a_nn form the main diagonal of A. The sum of the elements on the main diagonal of A is called the trace of A, and it is denoted by

A square matrix in which all the entries above (or below) the main diagonal are zero is called a lower (or an upper) triangular matrix. For example an upper triangular matrix has the form

$$A=\left(\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ 0& a_{22}& a_{23}& \dots & a_{2n}\\ 0& 0& a_{33}& \dots & a_{3n}\\ 0& 0& 0& \dots & a_{4n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & a_{nn}\end{array}\right)$$

A square matrix A = [a_ij]_n×n in which every entry off the main diagonal is zero, that is,

is called a diagonal matrix. The sum, the scalar product, and the product of diagonal matrices are again diagonal matrices. (The collection of all diagonal real n × n matrices is thus closed under these operations.)

Two matrices A = (a_ij) and B = (b_ij) are equal if every element of a_ij of A is equal to the correspondend element b_ij of B.

The set of al matrices m x n with elements in 𝕂 is denoted by 𝕄_m,n(𝕂) and the set of all square matricec of order n by M_n(𝕂). Obviously 𝕄_n,n(𝕂) = M_n(𝕂) and M₁(𝕂) = 𝕂, because a 1 x 1 matrix is nothing elese than an element of 𝕂

Sum of Matrices

By setting

∀(a_ij), (b_ij) ∈ 𝕄_m,n(𝕂), λ∈ 𝕂, is defined on 𝕄_m,n(𝕂) una struttura di 𝕂-spazio vettoriale. Lo zero è la matrice nulla.

If A = (a_ij) and B = (b_ij), λ∈ 𝕂, the matrices (a_ij) + (b_ij) and (λa_ij) are denoted by A + B and λA, respectively.

It is easy to verify that for the sum of matrices the commutative and associative properties hold:

Product of a matrix by a vector

Consideriamo il concetto importante di combinazione lineare di m vettori di dimensione n

questa combinazione lineare si può definire come il prodotto righe per colonne tra una matrice ed un vettore, quest'ultimo immaginato disposto come colonna.

$$\begin{array}{r}\left(\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1m}\\ a_{21}& a_{22}& \dots & a_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nm}\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_m\end{array}\right)=\left(\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{n1}\end{array}\right)x_1+\left(\begin{array}{c}{a}_{12}\\ a_{22}\\ ⋮\\ a_{n2}\end{array}\right)x_2+\left(\begin{array}{c}{a}_{1m}\\ a_{2m}\\ ⋮\\ a_{nm}\end{array}\right)x_m\\ ={\mathbf{a}}_1{x}_1+{\mathbf{a}}_2{x}_2+\dots +{\mathbf{a}}_m{x}_m=\left(\begin{array}{c}{a}_{11}{x}_1+a_{12}{x}_2+\dots +a_{1m}{x}_m\\ a_{21}{x}_1+a_{22}{x}_2+\dots +a_{2m}{x}_m\\ ⋮\\ a_{n1}{x}_1+a_{n2}{x}_2+\dots +a_{nm}{x}_m\end{array}\right)\end{array}$$

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