tailieunhanh - Handbook of mathematics for engineers and scienteists part 31

The segments forming a polygon are called the sides (or edges), and the points themselves are called the Venices of the polygon. Two sides sharing a vertex. as well as two successive vertices (the endpoints of the same edge), are said to be adjacent. A polygon can be self-intersecting, but the points of self-intersection should not be vertices (Fig. 3A3b). | 178 Algebra The integer r satisfying these two conditions is called the rank of the matrix A and is denoted by r rank A . Any nonzero rth-order minor of the matrix A is called its basic minor. The rows and the columns whose intersection yields its basic minor are called basic rows and basic columns of the matrix. The rank of a matrix is equal to the maximal number of its linearly independent rows columns . This implies that for any matrix the number of its linearly independent rows is equal to the number of its linearly independent columns. When calculating the rank of a matrix A one should pass from submatrices of a smaller size to those of a larger size. If at some step one finds a submatrix Ak of size k X k such that it has a nonzero kth-order determinant and the k 1 st-order determinants of all submatrices of size k 1 X k 1 containing Ak are equal to zero then it can be concluded that k is the rank of the matrix A. Properties of the rank of a matrix 1. For any matrices A and B of the same size the following inequality holds rank A B rank A rank B . 2. For a matrix A of size m Xn and a matrix B of size n X k the Sylvester inequality holds rank A rank B - n rank AB min rank A rank B . For a square matrix A of size n X n the value d n - rank A is called the defect of the matrix A and A is called a d-fold degenerate matrix. The rank of a nondegenerate square matrix A aj of size n X n is equal to n. Theorem on basic minor. Basic rows resp basic columns of a matrix are linearly independent. Any row resp. any column of a matrix is a linear combination of its basic rows resp. columns . . Expression of the determinant in terms of matrix entries. 1 . Consider a system of mutually distinct 31 32 . 3n with each 3i taking one of the values 1 2 . n. In this case the system 31 32 . 3n is called a permutation of the set 1 2 . n. If we interchange two elements in a given permutation 31 32 . 3n leaving the remaining n - 2 elements intact we obtain another permutation and