tailieunhanh - Handbook of mathematics for engineers and scienteists part 30

A polygon is a plane figure bounded by a closed broken line. a line obtained if one takes n distinct points such that no three successive points are collinear and draws a straight line segment between each of these points and its successor as well as between the last point and the first point (Fig. ). | . Matrices and Determinants 171 Remark. If a matrix is real . all its entries are real then the corresponding transpose and the adjoint matrix coincide. A square matrix A is said to be normal if A A AA . A normal matrix A is said to be unitary if A A AA I . A A-1 see Paragraph . . Trace of a matrix. The trace of a square matrix A aj of size n X n is the sum S of its diagonal entries n S Tr A aii. i 1 If A is a scalar and square matrices A and B has the same size then Tr A B Tr A Tr B Tr AA ATr A Tr AB Tr BA . Linear dependence of row vectors column vectors . A row vector column vector B is a linear combination of row vectors column vectors A1 . Ak if there exist scalars a1 . ak such that B a1A1 ak Ak. Row vectors column vectors A1 . Ak are said to be linearly dependent if there exist scalars a1 . ak a ak 0 such that a1 A1 ak Ak O where O is the zero row vector column vector . Row vectors column vectors A1 . Ak are said to be linearly independent if for any a1 . ak af ak 0 we have a1A1 akAk O. Theorem. Row vectors column vectors A1 . Ak are linearly dependent if and only if one of them is a linear combination of the others. . Inverse matrices. Let A be a square matrix of size n X n and let I be the unit matrix of the same size. A square matrix B of size n X n is called a right inverse of A if AB I .A square matrix C of size n X n is called a left inverse of A if CA I. If one of the matrices B or C exists then the other exists too and these two matrices coincide. In such a case the matrix A is said to be nondegenerate nonsingular . Theorem. A square matrix is nondegenerate if and only if its rows columns are linearly independent. Remark. Generally instead of the terms left inverse matrix and right inverse matrix the term inverse matrix is used with regard to the matrix B A 1 for a nondegenerate matrix A since AB BA I. Uniqueness Theorem. The matrix A-1 is the unique matrix satisfying the condition AA-1 A-1 A I for a given .