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 (a_{11}, a_{12}, ..., a_{1n}), (a_{21}, a_{22}, ..., 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 (a_{11}, a_{22}, ..., 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: A^{2} = AA, A^{3} = A^{2}A, ..., A^{0} = 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 (a_{11}) = |a_{11}|= a_{11} 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 (A^{n}) = (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! |