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 nn
n n n
n a a a a a a a a a 212222111211n D =It represents the n!
Algebra and items.These are all possible from different lines in different columns in the N element of the n nj j j a a a 2121 product.n nj j j a a a 2121symbols for )
(21)
1(n j j j τ-.When n j j j 21 is
even (odd) arrangement of the sign is positive (negative) .That is to say
n n
n nj j j j j j j j j a a a 21212121)
(n )1(D ∑-=
τ
.Here are
∑
n
j j j 21 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 !n so the n order determinant expansion there are !n ;
(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 n j j j 21 to I .How much is the inverse number and arrangement of 121j j j j n n -.
Analysis:If the number of the original array.1j earlier than 1j large number of 1τ, a number of smaller than 1j number is 1)1(τ--n , and the number of new arrangement of 121j j j j n n -1j earlier than 1j large number of as 1)1(τ--n ; similarly, a number of the original arrangement set before 2j than 2j a large number of 2τ, a number of smaller than 2j number is
2)2(τ--n , and the number of new arrangement of 2j earlier than 2j large number of as
2)2(τ--n ; followed by analogy, a number of the original arrangement set before k j than k j large number of k τ, t he number of new arrangement in k j before the number greater
than k j for ),,2,1()(n k k n k =--τ.
Solution: a number of set the original permutation of k j than k j large number of k τ, a new arrangement of
k j earlier than k j large number of a number of
),,2,1()(n k k n k =--τ. Because of I n =+++τττ 21 so Inverse number
arrangement
of
1
21j j j j n n - is
I n n k n n n n n
k k -=
+++-+++-+-=--=-=∑2
)1(211
)(]01)2()1[(])[(τττττ
2 basic theory
2.1 Properties of n order determinant
Property 1: Transpose, determinant. That is
nn
n n n n a a a a a a a a a
21
2222111211
=
nn
n n
n n a a a a a a a a a
212221212111
Property 2: a number multiplied by the determinant of a row is equal to the number is multiplied
by this determinant. That is nn
n n l l n
nn n n l l n a a a a a a a a a k a a a ka ka ka a a a
21ln 211121121ln 21
11211= Nature 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.nn
n n n n nn n n n n nn n n n n n a a a c c c a a a a a a b b b a a a a a a c b c b c b a a a 212111211
2121
1121121221111211+=+++ 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.
021
21ln
211121121
21ln 2111211==nn
n n in
i i l l n
nn n n in
i i l l n
a a a a a a a a a a a a k a a a a a a a a a a a a
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 ⎩⎨⎧≠=+++0,0,2211j
i D A a A a A a jn in j i j i Where ij A is a cofactor of element ij a . 2 Reduction theorem
B CA D A D
C
B A 1--=
3C
A C
B A =
4
B A AB =
5 the nonzero matrix K left by a row to another row determinant is the new block determinant and the original equal
2.3 results of several special determinant 1 triangular determinant
nn nn
n n a a a a a a a a a 2211222112110
00=（Triangular determinant ）
nn nn
n n a a a a a a a a a
221121
2221110
00=（Lower triangular determinant ）
2 diagonal determinant
nn nn
a a a a a a 221122110
000=
3 Symmetric and anti-symmetric determinant
nn
n n n
n a a a a a a a a a D 212222111211=To meet the )2,1,2,1(n j n i a a ji ij ===， D is called
symmetric determinant
0003
31332
312232111312 n n n n n
n a a a a a a a a a a a a D =To meet the )2,1,(n j i a a ji ij =-=, D is called
skew-symmetric determinant. If the order of n is odd then 0=D 4 Vandermonde determinant
∏≤<≤-----==n
i j j i n n n n n n
n n a a a a a a a a a a a a a a D 11
13121
12
23222
1321
)(1111。