Introduction into the matrix theory

Definition of matrix. By matrix A we mean a rectangular array of numbers

The matrix A consists of m horizontal n-tuples

(a₁₁, a₁₂, ..., a_1n), (a₂₁, a₂₂, ..., a_2n), ..., (a_m1, a_m2, ..., a_mn),

called rows of A, and of the n vertical m-tuples

called its columns. Note that the element a_ij called ij-entry, appears in the i-th row and the j-th column. We frequently denote such a matrix simply by A = (a_ij).

A matrix with m rows and n columns is said to be an m by n matrix, written m ´ n. The pair of numbers m and n is called the size of the matrix. Two matrices A and B are equal, written A = B, if they have the same size and if corresponding elements are equal.

It is possible to perform addition and multiplication of matrices.

Square matrix. A matrix with the same number of rows and as columns is called a square matrix. A square matrix with n rows and n columns is said to be of order n, and is called an n-square matrix. The main diagonal, or simply diagonal, of a square matrix A = (a_ij) consists of the numbers

(a₁₁, a₂₂, ..., a_nn).

Unit matrix. The n-square matrix with 1's along the main diagonal and 0'd elsewhere, e.g.

is called the unit or identity matrix and will be denoted by I. The unit matrix I plays the same role in matrix multiplication as the number 1 does in the usual multiplication of numbers. Specifically,

AI = IA = A.

for any square matrix A.

The square matrixes can be raised to a power. We can define powers of the square matrix A as following:

A² = AA, A³ = A²A, ..., A⁰ = I.

Inverse matrix. A square matrix A is said to be invertible if there exists a matrix B with the property that

AB = BA = I.

Such a matrix B is unique; it is called the inverse of A and is denoted by A^-1.

Determinants. To each n-square matrix A = (a_ij) we assign a specific number called the determinant of A, denoted by Det (A) or |A|.

The determinants of order one and two are defined as follows:

Det (a₁₁) = |a₁₁|= a₁₁

An important property of the determinant is given with the following theorem.

Theorem. For any two n-square matrices A and B we have

Det (AB) = (Det A) ´ (Det B).

It follows from this theorem that

Det (Aⁿ) = (Det A)ⁿ.

After this preliminary introduction we can go on to Fibonacci matrices. And we will begin from the simplest Fibonacci matrix, called Q-matrix. Follow us!