行列式的计算外文文献及翻译

1 The definition of determinant:n order determinant represented by the symbol It represents the n! Algebra and items.These are all possible from different lines in different columns in the N element of the product.symbols for .When is even (odd) arrangement of the sign is positive (negative) .That is to say .Here are said the summation of all n order.Definition of determinant by inverse number given, should grasp the following four:(1) The total number of n order permutation is so the n order determinant expansion there are ;(2) Each item is taken from different lines, product of n elements in different columns;(3) The premise in subscript according to the natural number sequence, the parity of the inverse number each item in front of the sign depends on the column index consisting of the arrangement of the evidence, the other half, half of them take the negative;(4) A number is the value of determinant.Example 1: if the permutation inversions number to .How much is the inverse number and arrangement of .Analysis:If the number of the original array.earlier than large number of, a number of smaller than number is , and the number of new arrangement of earlier than large number of as ; similarly, a number of the original arrangement set before than a large number of, a number of smaller than number is , and the number of new arrangement of earlier than large number of as ; followed by analogy, a number of the original arrangement set before than large number of , the number of new arrangement in before the number greater than for. Solution: a number of set the original permutation of than large number of, a new arrangement of earlier than large number of a number of . Because of so Inverse number arrangement of is 2 basic theory2.1 Properties of n order determinantProperty 1: Transpose, determinant. That is =Property 2: a number multiplied by the determinant of a row is equal to the number is multiplied by this determinant. That isNature 3: if a line is the two set of numbers and then the determinant equals two determinant and the determinant in addition to this, all with the original determinant of the corresponding line.Nature 4: if the determinant of two lines of the same so determinant is zero. The so-called two lines of the same is corresponding elements of two lines of equal.Nature 5: if the determinant of the two line is proportional to the determinant is zero.Nature 6: the same row to another row determinant factor.Nature 7: to wrap column position in two lines of the determinant of No.2.2 basic theory 1 Where is a cofactor of element .2 Reduction theorem3 45 the nonzero matrix K left by a row to another row determinant is the new block determinant and the original equal2.3 results of several special determinant1 triangular determinant（Triangular determinant）（Lower triangular determinant）2 diagonal determinant3 Symmetric and anti-symmetric determina