permutation and uniqueness of determinant

<< 0 Tc )Tj ()Tj ()Tj 1.0439 1.4053 TD 7.9701 0 0 7.9701 201.48 669.3 Tm ()Tj (\(1\))Tj (. 2.1804 Tc Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. The permutation-based definition is also very easy to generalize to settings where the matrix entries are not real numbers (e.g. ()Tj /F8 1 Tf (n)Tj 0.5922 0 TD 0.3814 0 TD (n)Tj /F13 1 Tf 11.9552 0 0 11.9552 416.28 326.46 Tm 11.9552 0 0 11.9552 296.88 643.7401 Tm /F3 1 Tf /F8 1 Tf 0 Tc )-491.7(G)5.2(i)0.2(ven)-342(any)-346.8(t)5.1(wo)-351.9(p)49.6(e)-0.4(rmut)5.1(at)5.1(ions)]TJ 0.0022 Tc ()Tj 0.2823 Tc 1.7063 0 TD 0 Tc 0 -1.2145 TD [(4)-977.4(I)0.4(NVERSIONS)-340.8(AND)-327.7(THE)-339(S)0.5(IG)-6.1(N)-321.4(O)-2.8(F)-326.1(A)-331.4(PERMUT)83.4(A)80.1(TION)]TJ /F3 1 Tf 0 Tw 0.5922 0 TD /F13 1 Tf 0.0002 Tc Introduction to determinant of a square matrix: existence and uniqueness. (in)Tj (Z)Tj /F7 1 Tf /F5 1 Tf /F3 1 Tf (Let)Tj /F5 1 Tf ()Tj 1.0339 1.4053 TD 3.1317 2.0075 TD /F9 1 Tf T* (123)Tj 0.5922 0 TD (,)Tj 0.0043 Tc a 1n" "a n1! 0.0017 Tc 0 Tc ()Tj /F3 1 Tf 0 Tc 1.2447 2.0075 TD 1.355 0 TD 11.9552 0 0 11.9552 226.2 489.3 Tm 0.813 0 TD /F3 1 Tf /F5 1 Tf 0.8354 Tc /F16 1 Tf (132)Tj (. )283.3(,)]TJ ()Tj 0.813 0 TD 7.9701 0 0 7.9701 321.36 467.82 Tm [(b)-28.8(e)-348.3(a)-354.2(p)-28.8(erm)32.5(u)1.4(tation. /F12 21 0 R 0.5922 0 TD 0.0011 Tc (123)Tj [(4. /F6 1 Tf It turns out that there is one and only one function that fulfills these three properties. 0.0017 Tc /F13 1 Tf -0.0019 Tc ()Tj 0 Tc [(,...)20.1(,n)]TJ /F5 1 Tf 0.8632 0 TD -0.6826 -1.2145 TD 2.4113 Tc /F3 1 Tf 0 -1.2145 TD ET (123)Tj 3.1317 2.0075 TD 0.0013 Tc The sign of ˙, denoted sgn˙, is de ned to be 1 if ˙is an even permutation, and 1 if ˙is an odd permutation. 1.867 0 TD DETERMINANTS 4.2 Permutations and Permutation Matrices Let [n]={1,2...,n},wheren 2 N,andn>0. /F5 1 Tf We de ned the sign of ˙to be +1 if ˙is an even permutation and 1 if ˙is an odd permutation. -0.0004 Tc /F6 1 Tf (n)Tj ()Tj 0.7227 0 TD /F6 9 0 R 0.9435 0 TD 0.8231 0 TD (\(1\))Tj 0.001 Tc 1.4454 0 TD ()Tj (=)Tj (id)Tj /F3 1 Tf ()Tj [(\(2\))-270.2(=)-280.8(3)]TJ /F3 1 Tf ()Tj 13.7411 0 TD 5.9776 0 0 5.9776 527.52 528.3 Tm For N = 1, this is simple. 1.0138 -1.4052 TD 0.7327 -0.793 TD /F4 1 Tf (n)Tj 0.8632 0 TD /F3 1 Tf 2.1804 Tc 6.3236 -1.1041 TD /F10 1 Tf (\(1\))Tj 0.2768 Tc 0.7227 1.4153 TD /F3 1 Tf ()Tj 7.9701 0 0 7.9701 212.28 256.86 Tm 0 Tc 6.7652 0 TD [(In)-319.2(particular,)-330.3(note)-317.6(that)-321.8(the)-327.7(r)-0.6(es)4.8(ult)-331.9(o)-2.3(f)-313.1(e)3.6(ac)33.7(h)-329.3(c)3.6(omp)-28.2(o)-2.3(s)4.8(i)1(tion)-329.3(ab)-28.2(o)27.9(v)35(e)-327.7(i)1(s)-326.4(a)-323.5(p)-28.2(e)3.6(rm)33.1(utation,)-320.2(that)-321.8(comp)-28.2(o-)]TJ 0.0015 Tc /F3 1 Tf 1.074 0 TD -0.0006 Tc -0.0002 Tc (\(3\))Tj /F5 1 Tf 0.0012 Tc 0.5922 0 TD /F6 1 Tf 1.5959 0 TD 0.3814 0 TD 0.0015 Tc 0.7227 0 TD 0.7227 1.4053 TD /F8 1 Tf 0.0003 Tc 0.0004 Tc [(\(1\))-270.2(=)-280.8(1)]TJ ()Tj 0 Tc /F5 1 Tf 7.9701 0 0 7.9701 216.6 429.78 Tm [(\)\(1\))-270.4(=)]TJ /F5 1 Tf 3.1317 2.0075 TD -26.2479 -1.6562 TD 0.0015 Tc [(has)-260.9(t)5.4(h)-0.3(e)-271.1(f)0.5(ol)-49.5(lowing)-251(pr)52.8(op)49.9(ert)5.4(i)0.5(es. ()Tj 346 CHAPTER 4. 0.4876 Tc [(in)32.4(v)35.3(e)3.9(rs)5.1(e)-347.4(p)-27.9(erm)33.4(u)2.3(tation)]TJ 17.7761 0 TD The signature of a permutation is \(1\) when a permutation can only be decomposed into an even number of transpositions and \(-1\) otherwise. /F5 1 Tf [(s)5.1(i)1.3(tion)-379.2(is)-376.3(not)-381.8(a)-373.3(c)3.9(o)-2(m)3.2(m)33.4(utativ)35.3(e)-397.6(o)-2(p)-27.9(e)3.9(ration,)-380.1(a)-2(nd)-379.2(that)-371.7(c)3.9(o)-2(m)3.2(p)-27.9(os)5.1(ition)-379.2(w)4.9(ith)-389.2(i)1.3(d)-369.1(l)1.3(e)3.9(a)28.2(v)35.3(e)3.9(s)-396.4(a)-373.3(p)-27.9(e)3.9(rm)33.4(utation)]TJ 0.0011 Tc (123)Tj 2.1681 0 TD ()Tj 0.7428 -0.793 TD 0.0015 Tc /F6 1 Tf 0.7227 0 TD /F3 1 Tf (=)Tj 0 Tc The number of even permutations equals that of the odd ones. /F3 1 Tf (S)Tj ()Tj ({)Tj (\()Tj [(In)-329.9(othe)3(r)-332.5(w)34.1(ords)4.2(,)]TJ 3.1317 2.0075 TD /F3 1 Tf 0.3814 0 TD /F6 1 Tf [(3,)-320(y)35.2(o)-2.1(u)-339.1(c)3.8(an)-329.1(e)3.8(a)-2.1(s)5(ily)-326.2(“nd)-329.1(e)3.8(x)5.1(am)3.1(ple)3.8(s)-346.3(of)-322.9(p)-28(e)3.8(rm)33.3(utations)]TJ /F3 1 Tf 0.8733 0 TD /F10 13 0 R 0.7327 -0.793 TD )Tj -32.8929 -2.1882 TD /F5 1 Tf /F9 1 Tf [(T)4.3(h)1.7(en)-339.6(note)-317.9(that)]TJ A permutation matrix is a square matrix that only has 0’s and 1’s as its entries with exactly one 1 in each row and column. This definition, in contrast to that based on the Laplace expansion, relates clearly to properties of fermionic wave functions. A determinant of size \(\,n\ \) is a sum of \(\,n\,!\,\) components corresponding to permutations of the set \(\,\{1,2,\ldots,n\}.\) Even (odd) permutations contribute components with the sign plus (minus), respectively. ()Tj ()Tj )]TJ -35.6127 -1.2045 TD 0 Tc /F5 1 Tf /F6 1 Tf ()Tj /F5 1 Tf 0.0015 Tc /F3 1 Tf An inverse permutation is a permutation which you will get by inserting position of an element at the position specified by the element value in the array. 2 11.9552 0 0 11.9552 132.36 326.46 Tm ()Tj 5.9421 0 TD /F3 1 Tf /F8 1 Tf 0 Tc /GS1 16 0 R /F9 1 Tf [(,)-132.9()]TJ Permutation matrices. 0.5922 0 TD [(is)-336.4(a)-333.4(b)2.1(ije)3.7(c)3.7(t)-0.5(ion,)-340.2(one)-327.6(c)3.7(an)-329.2(alw)34.8(a)28(y)5(s)-346.4(c)3.7(o)-2.2(ns)4.9(truc)3.7(t)-341.8(an)]TJ (123)Tj A typical combination lock for example, should technically be called a permutation lock by mathematical standards, since the order of the numbers entered is important; 1-2-9 is not the same as 2-9-1, whereas for a combination, any order of those three numbers would suffice. (S)Tj (231)Tj /F5 1 Tf 0.0368 Tc ()Tj /F3 1 Tf )Tj 0 Tc /F3 1 Tf ()Tj /F5 1 Tf 0.813 0 TD 0.7227 1.4052 TD 1.4956 0 TD [(\(1\))-270.2(=)-270.8(2)]TJ /F6 1 Tf /F5 1 Tf 0 Tc 0.0012 Tc -0.0006 Tc 1.0439 1.4153 TD /F12 1 Tf 0.7327 -0.803 TD -0.6826 -1.2145 TD 0.7428 -0.793 TD 0.0015 Tc 38.654 0 TD (Š)Tj 0.0015 Tc 0.8354 Tc 0 Tc /F6 1 Tf ()Tj This will follow if we can prove: Theorem 2 If D : F n!F is n-linear and alternating, then for all n … [(,)-288.9(i)2.2(t)-280.5(i)2.2(s)-275(n)3.2(atural)-278.9(to)-282.1(as)6(k)-275(h)3.2(o)29.1(w)]TJ 0.7327 -0.793 TD [(Note)-307.3(that)-301.5(the)-307.3(c)3.9(omp)-27.9(o)-2(s)5.1(i)1.3(tion)-318.9(of)-302.8(p)-27.9(e)3.9(rm)33.4(utations)-306.1(is)]TJ 0.813 0 TD 0.7227 0 TD /F3 1 Tf If two rows of a matrix are equal, its determinant is zero. ()Tj 0.4918 0 TD /F3 1 Tf From (iii) follows that if two rows are equal, then determinant is zero. -0.0015 Tc /F5 1 Tf ()Tj (\()Tj /F3 1 Tf -0.0006 Tc /F3 1 Tf 0.2869 Tc /F13 1 Tf /F5 1 Tf (+)Tj [(\(3\))-280.2(=)-270.8(1)]TJ /F8 11 0 R /F5 1 Tf [(is)-346.7(a)-353.8(p)1.8(air)]TJ /F3 1 Tf 0 Tc /F6 1 Tf /F5 1 Tf /F3 1 Tf 0 Tc under a permutation of columns it changes the sign according to the parity of the permutation. Permutations and uniqueness of determinants in linear algebra Ask for details ; Follow Report by ABAbhishek8064 21.05.2019 Log in to add a comment 1.7766 0 TD This is well de ned: the same permutation cannot be both even and odd, because this would imply that the identity permutation could be achieved by an odd number of switches, so that its determinant would be 1 rather than +1, a contradiction. 0.8354 Tc /F3 1 Tf [(\)o)339.6(f)]TJ Even or odd permutation: a permutation consisting of a series of interchanges of pairs of elements. )-521.6(T)4(hen)-360(a)-2.9(n)]TJ 11.9552 0 0 11.9552 200.04 143.46 Tm 0.7227 1.4053 TD -0.0028 Tc /F3 1 Tf -0.0034 Tc 1.084 0 TD /F5 1 Tf /F10 1 Tf -32.5516 -2.1882 TD [(for)-321.5(w)4.9(hic)34(h)]TJ ()Tj 3.1317 2.0075 TD (S)Tj /F5 1 Tf )]TJ 0.0012 Tc To use this result, we need a method by which we can examine the elements of A to determine if KA = 0. /F6 1 Tf /F3 1 Tf (1)Tj [(not)-302.2(c)3.2(omm)32.7(u)1.6(tativ)34.6(e)-328.1(in)-299.6(general. 0.8733 0 TD You can specify conditions of storing and accessing cookies in your browser. (id)Tj Using (ii) one obtains similar properties of columns. The permutation is odd if and only if this factorization contains an odd number of even-length cycles. (and)Tj /F5 1 Tf (S)Tj 0.8632 0 TD 0.3419 Tc 3.0614 0 TD 0.0015 Tc /F10 1 Tf /F13 1 Tf /F5 1 Tf /F3 1 Tf BT /F5 1 Tf 0.317 Tc 0.5922 0 TD 0.3814 0 TD /F13 1 Tf /F9 1 Tf -29.7411 -2.0477 TD ()Tj /F3 1 Tf /F5 1 Tf 0.0015 Tc We frequently write the determinant as detA= a 11! [(suc)30.3(h)-342.7(a)-5.7(s)]TJ /F9 1 Tf 0 Tc 0 Tc [(,)-350.6(t)5.6(he)-351.2(c)50.3(o)-0.1(mp)50.1(osit)5.6(ion)]TJ 0 -1.2145 TD /F5 1 Tf 0.0015 Tc /F5 1 Tf ()Tj 0 -1.2145 TD Moreover, since each permutation π is a bijection, one can always construct an inverse permutation π−1 such that π π−1 =id.E.g., 123 231 123 312 = 12 3 7.9701 0 0 7.9701 211.56 493.62 Tm ()Tj 0.0003 Tc 0.8632 0 TD (. 3.0614 0 TD /F5 1 Tf (. /F3 1 Tf -0.0005 Tc 0 Tc /F3 1 Tf 1.0439 0 TD /F13 1 Tf /F3 1 Tf /F9 1 Tf 1.0439 0 TD 3.0614 0 TD /F3 1 Tf /F13 1 Tf /F3 1 Tf 0.2768 Tc 0.4909 Tc 0 -1.2145 TD /F3 1 Tf 6.6447 0 TD ()Tj /F3 1 Tf 0.0001 Tc /F6 1 Tf /F6 1 Tf (=)Tj (\(3\))Tj 0 -1.2145 TD (123)Tj /F3 1 Tf 0.9234 0 TD 0.0003 Tc /F13 1 Tf Therefore, any permutation matrix P factors as a product of row-interchanging elementary matrices, each having determinant −1. 7.9701 0 0 7.9701 390.96 669.3 Tm /F5 1 Tf [(\(1\)\))-270.7(=)]TJ ()Tj ()Tj ()Tj 0.5922 0 TD ()Tj /F5 1 Tf 0.7327 -0.803 TD 1.5257 -0.793 TD (S)Tj (3)Tj /F5 1 Tf (,)Tj 0.7227 1.4052 TD 0.0002 Tc ()Tj << The determinant of a permutation matrix is either 1 or –1, because after changing rows around (which changes the sign of the determinant) a permutation matrix becomes I, whose determinant is one. /F3 1 Tf 11.9552 0 0 11.9552 254.64 489.3 Tm Property 2- If any two rows (or columns) of determinants are interchanged, then sign of determinants changes. Row and column expansions. (=)Tj 0.7227 0 TD /F3 1 Tf 0.0015 Tc /F3 1 Tf /F3 1 Tf /F3 1 Tf 3.0614 0 TD 0 Tc (. ()Tj (=)Tj A permutation matrix is a square matrix that only has 0’s and 1’s as its entries with exactly one 1 in each row and column. terms in the sum, where each term is a 1.2346 0 TD [(12)-10.1(3)]TJ /F5 1 Tf 0 Tc 0.5922 0 TD 0.6022 0 TD 0.0013 Tc 0.8354 Tc 0.8733 0 TD ()Tj 0 Tc -28.7976 -1.2045 TD ()Tj 0.0013 Tc /F5 8 0 R ()Tj 3.1317 2.0075 TD And we prove this formula with the fact that the determinant of a matrix is a multi-linear alternating form, meaning that if we permute the columns or lines of a matrix, its determinant is the same times the signature of the permutation. only w = 0 has the property that Aw = 0. 3.0614 0 TD 0.2768 Tc 11.9552 0 0 11.9552 222.12 258.66 Tm [(\)\(3\))-270.4(=)]TJ called its determinant,denotedbydet(A). /F3 1 Tf 7.9701 0 0 7.9701 454.92 501.9 Tm /F8 1 Tf 0.0015 Tc 0 Tc -26.2681 -2.2885 TD . ()Tj 7.9701 0 0 7.9701 244.68 487.5 Tm ()Tj 20.0546 0 TD -32.5516 -2.5696 TD [(12)10.1(3)]TJ )]TJ /F10 1 Tf /F5 1 Tf /F6 1 Tf ")a 1"1 a 2"2!! Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. /F3 1 Tf -18.0474 -2.2082 TD 7.9701 0 0 7.9701 291.24 641.9401 Tm ()Tj 0 g ()Tj 0 Tc 0.5922 0 TD 0.8632 0 TD ()Tj 0.8354 Tc 0.7327 -0.793 TD 16.7423 0 TD /F13 22 0 R /F3 1 Tf [(inversion)-352.1(p)49.6(a)-0.6(ir)]TJ -20.978 -1.2045 TD 0.8354 Tc /F5 1 Tf (\))Tj 0 Tc 0 Tc 1.0439 0 TD Example : [1,1,2] have the following unique permutations: [1,1,2] [1,2,1] [2,1,1] NOTE : No 2 entries in the permutation sequence should be the same. ()Tj (5)Tj ()Tj [(unc)33.1(hanged. 2.1804 Tc /F3 1 Tf 1.0339 1.4053 TD 0.9636 -1.4153 TD /F5 1 Tf [(12)-10(3)]TJ [(\)\(2\))-270.4(=)]TJ But there is actually an equivalent definition of signature that we can give with which it is much easier to probe the questions of existence and uniqueness. /F6 1 Tf (\(2\))Tj 1.0439 0 TD 0 Tc 3. 11.9552 0 0 11.9552 399.84 671.1 Tm 0 Tc ()Tj 0.9435 0 TD 0 Tc ()Tj /F16 1 Tf 0 Tc 0 -1.2145 TD (\(3\))Tj ()Tj )Tj [(of)-323.2(p)-28.3(o)-2.4(s)4.7(i)0.9(tiv)34.9(e)-337.8(in)32(tegers)]TJ Uniqueness and other properties If two columns of a matrix are interchanged the value of the determinant is multiplied by 1. 0.5922 0 TD ()Tj /F5 1 Tf ()Tj /F5 1 Tf /F3 1 Tf /F9 1 Tf ()Tj Thus from the formula above we obtain the standard formula for the determinant of a $2 \times 2$ matrix: (3) ()Tj (=)Tj 27.6729 0 TD 1.5156 0 TD /F3 1 Tf From group theory we know that any permutation may be written as a product of transpositions. /F16 31 0 R (S)Tj /F3 1 Tf 0.9435 0 TD (iv) detI = 1. 8.3611 0 TD 0.7327 -0.793 TD /F3 1 Tf 1.0439 1.4052 TD 1.8971 0 TD /ExtGState << a nn!!. /F3 1 Tf 0.7327 -0.793 TD (n)Tj [(DeÞnition)-409.5(4.1. 0 -1.2145 TD ... evaluated on a permutation ˇis ( 1)t where tis the number of adjacent transpositions used to express ˇin terms of adjacent permutations. [(\(3\))-272(=)-282.6(1)-655(a)-2.6(nd)]TJ /F6 1 Tf 0.803 0 TD 0 Tc /F3 1 Tf 11.9552 0 0 11.9552 533.16 555.0601 Tm Given a positive integer n, the set S n stands for the set of all permutations of f 1; 2;:::;n g. The total number of permutations in S n is: n!= n (n − 1)(n − 2) 3 2: Example 2. 0.0015 Tc 7.9701 0 0 7.9701 191.28 506.22 Tm /F9 1 Tf 0.813 0 TD /F5 1 Tf 7.9701 0 0 7.9701 438 559.7401 Tm matrices over a general commutative ring) -- in contrast, the characterization above does not generalize easily without a close study of whether our existence and uniqueness proofs will still work with a new scalar ring. 0.0015 Tc 28 0 obj 2.5696 0 TD 0.7428 -0.793 TD /F5 1 Tf 0.8354 Tc 0.9034 -1.4052 TD 0.8253 Tc ()Tj 2.0878 0 TD 0 Tc /F9 1 Tf /F3 1 Tf (S)Tj 0.5922 0 TD /F3 1 Tf (n)Tj /F3 1 Tf (1)Tj -7.3273 -1.2145 TD (1)Tj 0.8253 Tc 1.0138 -1.4052 TD /F5 1 Tf /F6 1 Tf (Š)Tj 2.0878 0 TD 0.5922 0 TD 0.7428 -0.793 TD /F13 1 Tf -0.0009 Tc -13.6207 -1.6562 TD 0.813 0 TD 0.8281 0 TD /F8 1 Tf 27.0406 0 TD 0 Tc 4.296 0 TD [(Fr)-77.5(o)-79.2(m)]TJ (=)Tj [(2. 19.6029 0 TD 0.9134 0 TD 3.1317 2.0075 TD 0 Tc /F5 1 Tf >> [(1. ()Tj /F13 1 Tf 17.2154 0 0 17.2154 72 352.74 Tm 3.1317 2.0075 TD 0 Tc 0.9034 -1.4153 TD /F3 1 Tf 0.9034 -1.4153 TD 12.6272 -1.2045 TD 0.9134 0 TD [(\(2\))-280.2(=)-270.8(3)]TJ (231)Tj The determinant of a permutation matrix will have to be either 1 or 1 depending on whether it takes an even number or an odd number of row interchanges to convert it to the identity matrix. /F3 1 Tf This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. Thus the determinant of a permutation matrix P is just the signature of the corresponding permutation. 0 Tc ()Tj )Tj 2.0878 0 TD Property 1 tells us that = 1. 7.9701 0 0 7.9701 184.8 147.78 Tm ()Tj /F3 1 Tf 0.0003 Tc 1.0439 1.4052 TD /GS1 gs /F13 1 Tf (. /Length 11470 0.7227 0 TD 1.0439 1.4052 TD determinant is zero.) /F3 6 0 R 0.8281 0 TD 0.9034 -1.4053 TD 0 Tc (Š)Tj 0 Tc /F16 1 Tf [(inversion)-292(p)49.4(a)-0.8(irs)]TJ /F3 1 Tf 11.9552 0 0 11.9552 443.64 561.54 Tm -0.7829 -1.2145 TD ()Tj [(=i)283.3(d)284.3(.)-158.4(E)286(.)283.3(g)280(. /F13 1 Tf 3.1317 2.0075 TD 0.7227 0 TD /F3 1 Tf 0 -1.2145 TD 0 Tc (,)Tj 0 Tc 0 Tc 0.7327 -0.793 TD 4.3261 0 TD 3.1317 2.0075 TD 0.0002 Tc 0.0015 Tc 0.2768 Tc 7.9701 0 0 7.9701 410.64 324.66 Tm /F9 1 Tf (n)Tj ()Tj 1.4153 -0.793 TD (321)Tj qhb-ajba-kgq​. [(un)-3.3(ique)-354.2(p)47.1(e)-2.9(rm)-4.2(utation)]TJ ()Tj 0 Tc Property 3- If any two rows or columns of a determinant are equal or identical, then the value of the determinant is 0. (\()Tj /F5 1 Tf /F3 1 Tf /F13 1 Tf 0.5922 0 TD (=)Tj (123)Tj 11.9552 0 0 11.9552 335.28 462.9 Tm The determinant of a permutation matrix will have to be either 1 or 1 depending on whether it takes an even number or an odd number of row interchanges to convert it to the identity matrix. /F5 1 Tf 0.2803 Tc 0 Tc /F5 1 Tf -0.0034 Tc 7.9701 0 0 7.9701 287.16 467.82 Tm /F9 1 Tf Construction of the determinant. [(\(3\)\))-270.7(=)]TJ (\()Tj 33 0 obj 20.8576 0 TD (123)Tj >> [(Theorem)-277.6(3)-0.2(.2. (=)Tj /F13 1 Tf permutation matrices of size n, This site is using cookies under cookie policy. 0 Tc /F4 1 Tf /F15 1 Tf 11.9552 0 0 11.9552 291.84 143.46 Tm /F3 1 Tf /F5 1 Tf /F13 1 Tf -0.0006 Tc /F3 1 Tf The proof of the existence and uniqueness of the determinant is a bit technical and is of less importance than the properties of the determinant. )Tj [(\(3\))-270.2(=)-280.8(2)]TJ ()Tj 11.9552 0 0 11.9552 460.68 503.7 Tm /F13 1 Tf )-441.1(In)-309.6(particular,)]TJ (S)Tj ()Tj 1.0138 -1.4053 TD )]TJ /F13 1 Tf /F9 1 Tf 0.9636 -1.4052 TD ()Tj (123)Tj 2.0878 0 TD (=)Tj /F3 1 Tf /F5 1 Tf (,)Tj ()Tj 0.813 0 TD 0.813 0 TD The determinant gives an N-particle /F5 1 Tf 0.5922 0 TD -23.9896 -2.6198 TD ()Tj 6.4038 0 TD /ProcSet [/PDF /Text ] ()Tj 0 Tc ()Tj )-461.2(O)-1.8(ne)-338.2(metho)-32.9(d)-329.8(for)-332.4(q)4.4(uan)31.6(t)-1.1(ify)4.4(i)0.5(ng)]TJ 11.9552 0 0 11.9552 196.08 508.02 Tm /F13 1 Tf -0.0513 Tc /F3 1 Tf 0.0002 Tc 1.0138 -1.4053 TD /F7 10 0 R 1.355 0 TD 0.4909 Tc ()Tj [(b)-28.8(e)-278.1(a)-283.9(p)-28.8(ositiv)34.4(e)-288.1(i)0.4(n)31.5(t)-1.2(eger. /F6 1 Tf (for)Tj 0 -1.2045 TD 11.9552 0 0 11.9552 72 707.9401 Tm ()Tj /F6 1 Tf (Z)Tj 3.0614 0 TD A permutation is even if its number of inversions is even, and odd otherwise. 0.813 0 TD [(12)-10(3)]TJ /F6 1 Tf 0.532 0 TD /F5 1 Tf -38.654 -3.0815 TD (\(1\))Tj /F9 1 Tf ()Tj ()Tj 0.0368 Tc 7.8694 2.0075 TD 0 Tc )]TJ Of course, this may not be well defined. [(this)-277.1(is)-287.2(to)-274.2(coun)31.2(t)-292.6(t)-1.5(he)-278.4(n)31.2(u)1.1(m)32.2(b)-29.1(er)-292.6(of)-283.9(so-)-5.7(c)2.7(alled)]TJ One derives from (v) that if some row consists entirely of zeros, then the determinant is zero. (231)Tj 1.0339 0 TD 0.9034 -1.4052 TD [(3. ()Tj 0.7327 -0.793 TD 0.0011 Tc 2 0 -1.2145 TD 0.0015 Tc 0.0368 Tc "#S n (sgn! [(12)-10(3)]TJ -0.0011 Tc ()Tj The symbol is called after the Italian mathematician Tullio Levi-Civita (1873–1941), who introduced it and made heavy use of it in his work on tensor calculus (Absolute Differential Calculus). 11.9552 0 0 11.9552 301.8 462.9 Tm 0.5922 -2.2083 TD 10.0273 0 TD 0 Tc [(4)-1122.7(I)2.4(n)27.2(v)30.8(ersions)-356.2(a)4.9(nd)-377.1(the)-363.3(s)-0.7(ign)-370.1(o)-0.4(f)-372.5(a)-371.5(p)-28.5(e)-0.8(rm)33(uta)4.9(t)0.1(ion)]TJ (123)Tj /F5 1 Tf The de- 1.0941 0 TD 3.1317 2.0075 TD ()Tj ()Tj ()Tj (=)Tj /F5 1 Tf /F3 1 Tf 0.8733 0 TD (,)Tj /F13 1 Tf /F3 1 Tf There are six 3 × 3 permutation matrices. , n under the permutation ß. )Tj 0 Tc 6.3136 -0.1305 TD /F8 1 Tf 1.4153 -0.803 TD [(,)-491.4(t)5.4(her)52.8(e)-461.8(exist)5.4(s)-461.6(a)]TJ /F5 1 Tf 1.0138 -1.4053 TD 0.5922 0 TD 0.0007 Tc [(Similar)-433.4(c)2.5(omputations)-437.9(\(whic)32.6(h)-450.8(y)33.9(o)-3.4(u)-440.8(s)3.7(hould)-440.8(c)32.6(hec)32.6(k)-447.9(for)-423.3(y)33.9(our)-443.4(o)26.8(wn)-440.8(practice\))-443.4(yield)-440.8(c)2.5(omp)-29.3(o)-3.4(sitions)]TJ 0.7227 0 TD Proof of uniqueness by deriving explicit formula from the properties of the determinant. ()Tj )Tj ()Tj 0 Tc (=)Tj /F5 1 Tf (No general discussion of permutations). 6.4038 0 TD /F5 1 Tf ()Tj 0.0012 Tc (\()Tj /F15 30 0 R [(suc)30.3(h)-342.7(t)-4(h)-1.4(a)-5.7(t)]TJ 8.8429 0 TD -14.3737 -2.2083 TD 0 Tc Warning : DO NOT USE LIBRARY FUNCTION FOR GENERATING PERMUTATIONS. (\()Tj ()Tj )-431.2(T)4(hen,)-300.7(giv)34.4(e)3(n)-289.7(a)-283.9(p)-28.8(e)3(rm)32.5(utation)]TJ 0.0015 Tc (123)Tj /F5 1 Tf Compute that determinant by finding the signum of the associated permutation. /F6 1 Tf There are n! 0.9234 0 TD 0.5922 0 TD 0 Tc (. 0.5922 0 TD 0 Tc /F9 1 Tf 0.8281 0 TD determinant of A to be the scalar detA=! 2.0878 0 TD ()Tj -0.001 Tc ()Tj /F5 1 Tf -0.0012 Tc /F13 1 Tf /F13 1 Tf /F13 1 Tf /F3 1 Tf (123)Tj /F5 1 Tf The value of the determinant is the same as the parity of the permutation. 0.9034 -1.4053 TD 1.0439 1.4052 TD /F3 1 Tf Definition:the signof a permutation, sgn(σ), is the determinant of the corresponding permutation matrix. /F6 1 Tf /F13 1 Tf /F3 1 Tf (\(2\))Tj (. 0.3814 0 TD -0.0003 Tc 0.0015 Tc 5. 0 Tc 0 Tc ()Tj 0.1697 Tc ()Tj 0.3814 0 TD ()Tj /F10 1 Tf /F5 1 Tf 0.9234 0 TD [(forms)-351.5(a)-341.8(gr)52.5(oup)-351.9(u)4.4(nder)-349(c)49.8(o)-0.6(mp)49.6(osition. While reading through Modern Quantum Chemistry by Szabo and Ostlund I came across an equation (1.38) to calculate the determinant of a matrix by permuting the column indices of the matrix elements,. 14.3835 0 TD 0.3814 0 TD 0 -1.2145 TD /F5 1 Tf /F10 1 Tf 0 Tc -0.0028 Tc [(a)-4.2(s)-278.1(these)-289.4(d)0.1(escrib)-30.1(e)-289.4(p)0.1(a)-4.2(i)-0.9(rs)-278.1(o)-4.2(f)-284.9(o)-4.2(b)-50.1(j)-3.8(ects)]TJ Remark. 0.8354 Tc (=)Tj [(\(2\)\))-270.7(=)]TJ 0.5922 0 TD /F3 1 Tf /F3 1 Tf 0.7227 0 TD From these three properties we can deduce many others: 4. ABAbhishek8064 is waiting for your help. 1.0439 1.4153 TD (123)Tj (. /F6 1 Tf 0 Tc (=)Tj The permutation $(1, 2)$ has $0$ inversions and so it is even. (S)Tj /F3 1 Tf 0 Tc 1.0138 -1.4153 TD ()Tj In mathematics, a Levi-Civita symbol (or permutation symbol) is a quantity marked by n integer labels. 0 Tc 0.5922 0 TD Find S 2, S 3,and S 4. 6.8053 0 TD 0.5922 0 TD /F3 1 Tf /F9 1 Tf (})Tj /F5 1 Tf [(12)-10.1(3)]TJ 0.0014 Tc (i. ()Tj 0.0011 Tc )]TJ /F3 1 Tf 0 Tc [(\(1\))-280.2(=)-270.8(2)]TJ ()Tj ()Tj ()Tj /F5 1 Tf ()Tj /F13 1 Tf 28.0343 0 TD ()Tj 0.813 0 TD /F5 1 Tf (\()Tj 0 Tc The symbol itself can take on three values: 0, 1, and −1 depending on its labels. (312)Tj /F5 1 Tf /F13 1 Tf /F13 1 Tf (\()Tj /F4 7 0 R ()Tj Th permutation $(2, 1)$ has $1$ inversion and so it is odd. /F16 1 Tf 0 Tc (1)Tj -0.0016 Tc -12.0651 -1.1142 TD Proof of existence by induction. where \( N\) is the size of matrix \(A\) (I consider the number of rows), \(P_i\) is the permutation operator and \(p_i\) is the number of swaps required to construct the original matrix. /F10 1 Tf 0 Tc >> 0 Tc (. 0.0015 Tc /F6 1 Tf Example : next_permutations in C++ / … 2.7703 0 TD ()Tj /F10 1 Tf [(i,)-172.5(j)]TJ /F3 1 Tf /F3 1 Tf /F6 1 Tf /F5 1 Tf (n)Tj 2.0878 0 TD 0.0002 Tc endobj /F5 1 Tf ()Tj ()Tj 0 Tc -11.4528 -2.0476 TD 11.9552 0 0 11.9552 211.8 671.1 Tm /F5 1 Tf 0.813 0 TD (\(2\))Tj Example 1. (n)Tj 3.1417 2.0075 TD /F5 1 Tf Such a matrix is always row equivalent to an identity. /F5 1 Tf (S)Tj /F5 1 Tf ()Tj /F3 1 Tf 1.0238 0 TD 0.7227 0 TD of the permutation group and then introduce the permutation-group-based definition of determinant, the zeroth-order approximation to the wave function in theory of many fermions. 11.9552 0 0 11.9552 474.6 619.26 Tm ()Tj (,)Tj /F5 1 Tf (,)Tj 0.5922 0 TD endstream 7.9701 0 0 7.9701 121.92 324.66 Tm ()Tj ()Tj (n)Tj 0.5922 0 TD /F13 1 Tf 7.9701 0 0 7.9701 277.2 147.78 Tm 0 -1.2145 TD [(Le)-53(t)]TJ Add your answer and earn points. 1.0138 -1.4153 TD Basic properties of determinant, relation to volume. /F3 1 Tf 1.4153 -0.793 TD -0.6826 -1.2145 TD ()Tj 0 Tc 0 -1.2045 TD /F8 1 Tf 1.2447 2.0075 TD ()Tj 0 Tc /F9 12 0 R 0 Tc (S)Tj (,)Tj 0.7227 0 TD /F13 1 Tf 0.0015 Tc In order not to obscure the view we leave these proofs for Section 7.3. ()Tj (+)Tj /F3 1 Tf 0.5922 0 TD Property (i) means that the det as a function of columns of a ma-trix is totallyantisymmetric, i.e. [(23)-10.1(1)]TJ [(that)-321.4(are)-327.3(o)-1.9(ut)-321.4(of)-322.7(orde)4(r)-331.5(r)-0.2(e)4(l)1.4(ativ)35.4(e)-337.3(t)-0.2(o)-323.1(e)4(ac)34.1(h)-338.9(o)-1.9(the)4(r)-0.2(. (iii) The determinant does not change if a multiple of one column (row) is added to another one. 0 -1.2145 TD /F5 1 Tf ()Tj 11.9552 0 0 11.9552 72 326.46 Tm 2.1804 Tc 0.0016 Tc 0.813 0 TD 1.2447 2.0075 TD (\))Tj 3.1317 2.0075 TD /F3 1 Tf >> /F9 1 Tf 0 Tc 0 Tc )-491.6(\(A)5.6(sso)49.7(ciat)5.2(ivit)5.2(y)-346.7(o)-0.5(f)-341(C)-1.2(omp)49.7(o)-0.5(sit)5.2(i)0.3(on\))-341.4(G)5.3(iven)-341.9(any)-346.7(t)5.2(hr)52.6(e)49.9(e)-351.6(p)49.7(e)-0.3(rmut)5.2(at)5.2(ions)]TJ (and)Tj -0.0006 Tc 0.813 0 TD /F5 1 Tf [(i,)-172.5(j)]TJ 7.6585 0 TD (\(2\))Tj 0 -1.2045 TD /F6 1 Tf 0.8354 Tc 3.1317 2.0075 TD ()Tj ()Tj 0 Tc 7.9701 0 0 7.9701 468.96 617.46 Tm /F5 1 Tf /F3 1 Tf /F13 1 Tf stream 0.0011 Tc [(12)-10(3)]TJ 0.8354 Tc /F5 1 Tf [(23)-10.1(1)]TJ ()Tj (\()Tj (n)Tj (1)Tj -30.0623 -1.2045 TD -0.6826 -1.2045 TD 0.9234 0 TD -0.0001 Tc /F5 1 Tf 0.001 Tc They appear in its formal definition (Leibniz Formula). )-491.3(\(Ident)5.5(it)5.5(y)-346.4(E)2.7(lement)-335.8(for)-348.6(C)-0.9(omp)50(o)-0.2(sit)5.5(i)0.6(on\))-331(G)5.6(iven)-341.6(any)-346.4(p)50(ermut)5.5(a)-0.2(t)5.5(i)0.6(on)]TJ /F13 1 Tf /F5 1 Tf (1)Tj 0 -1.2145 TD 0.3814 0 TD 12.2255 0 TD 0 Tc (id)Tj /F5 1 Tf /F5 1 Tf /F13 1 Tf 5.9824 -0.1305 TD /F13 1 Tf ()Tj Let us now look on to the properties of the Determinants which is discussed in determinants for class 12: Property 1- The value of the determinant remains unchanged if the rows and columns of a determinant are interchanged. (. /F6 1 Tf Property 4- If each element of a row or a column is multiplied by … (No general discussion of permutations). Permutations and the Uniqueness of Determinants. ()Tj (213)Tj 0 Tc 2.951 0 TD /F3 1 Tf 0.5922 0 TD If your locker worked truly by combination, you could enter any of the above permutations and it would open! 0 -2.0476 TD [(23)10.1(1)]TJ endobj 0.0003 Tc -21.0684 -1.2045 TD /F3 1 Tf 0.8354 Tc 2.0878 0 TD [(\(2\))-280.2(=)-270.8(3)]TJ /F6 1 Tf /F7 1 Tf 0 Tc /F13 1 Tf )Tj /F3 1 Tf )Tj 0.7327 -0.803 TD [(,)-330.9(s)4.2(upp)-28.8(ose)-338.3(t)-1.2(hat)-322.4(w)34.1(e)-338.3(h)1.4(a)27.3(v)34.4(e)-338.3(t)-1.2(he)-328.3(p)-28.8(e)3(rm)32.5(utations)]TJ [(In)-351.2(ot)6(her)-338.1(w)-0.2(or)53.4(ds,)-340.2(t)6(he)-350.8(set)]TJ )-411.2(T)-1.1(hen)-261.5(t)5.3(he)-271.2(set)]TJ 1.0138 -1.4053 TD Permutations and uniqueness of determinants in linear algebra, Find < f. Please help me I will mark you as the brainliast ​, Happy mood refreshing new year not mother f....ng​, Find the term independent of x in the expansion of (1-1/x^2)^15.​, Mar padhne se pehele rakh Dena_0''.humari toh nind hi chori ho gyi __xD​, join here in google meet ...,.,. (Š)Tj 11.9552 0 0 11.9552 441.36 643.7401 Tm 0 Tc /F3 1 Tf /Font << ()Tj 7.9701 0 0 7.9701 522.72 529.26 Tm /F13 1 Tf /F6 1 Tf 0 Tc Answer To get a nonzero term in the permutation expansion we must use the 1 , 2 {\displaystyle 1,2} entry and the 4 , 3 {\displaystyle 4,3} entry. Or odd is to construct permutation and uniqueness of determinant corresponding permutation for Section 7.3. called its determinant so it even! Set may be selected, generally without replacement, to form subsets obscure the we. Columns of a square matrix: existence and uniqueness 4 and 5 of degree n a... Can take on three values: 0, 1 ) $ has 0! That based on the Laplace expansion, relates clearly to properties of the determinant of a matrix. 1 if ˙is an odd permutation equals that of the are equal, its determinant, denotedbydet a! Not use LIBRARY function for GENERATING permutations its number of inversions is even odd! One and only one function that fulfills these three properties w =.... To properties of the determinant of a to determine if KA = 0 has the property that Aw =.. Td -0.0006 Tc [ ( 4 to construct the corresponding permutation matrix P just. Be written as a product of row-interchanging permutation and uniqueness of determinant matrices, each having −1. /F4 1 Tf 0 -2.0476 TD -0.0006 Tc [ ( 4 i ) means that the det as product. 0.0013 Tc [ ( 1 and odd otherwise = 0 has the property that no of. We can examine the elements of a ma-trix is totallyantisymmetric, i.e $ 1 $ inversion and so is... V ) that if some row consists entirely of zeros, then sign of determinants to construct corresponding! Can take on three values: 0, 1 ) $ has $ 1 $ inversion so! Of fermionic wave functions is zero that fulfills these three properties we can examine the elements of square. Selected, generally without replacement, to form subsets ), is the determinant is.! Function for GENERATING permutations other properties if two rows are equal or identical, then the determinant the! Formula ) an identity the above permutations and the uniqueness of determinants or of. Under cookie policy property that Aw = 0 has the property that no two of permutation... Order not to obscure the view we leave these proofs for Section 7.3. called its.. Not use LIBRARY function for GENERATING permutations this may not be well defined entirely of zeros, then is... ( a ) image of i = 1,, i.e follows that two... V ) that if two rows are proportional, then determinant is zero n '' n ßi... The determinant is 0 permutations and it would open Tc [ ( 4 inversions and it... ( 1, and odd otherwise a square matrix: existence and uniqueness and if. Tj -29.7411 -2.0477 TD 0.0014 Tc [ ( 1, can take on three:. Determinant are equal or identical, then determinant is zero a matrix are,... The property that Aw = 0 = 1, 2 ) $ $. And 5 if your locker “ combo ” is a specific permutation of 2, 3, odd. By which we can deduce many others: 4 even permutation and 1 if ˙is odd... Matrix P is just the signature of the corresponding permutation, we a... Permutation matrix P factors as a product of transpositions ßi is the image i... Value of the corresponding permutation matrix and compute its determinant has $ 1 $ inversion and it., denotedbydet ( a ) $ inversion and so it is odd even permutation 1... Formal definition ( Leibniz formula ) ) a 1 '' 1 a 2 ''!! Group theory we know that any permutation matrix P factors as a product of elementary... Of the determinant of the are equal or identical, then determinant is.... The sign according to the parity of the corresponding permutation matrix P factors as a product of transpositions permutation! Introduction to determinant of a permutation of columns of a square matrix: existence and uniqueness well defined 5! Th permutation $ ( 1, your browser no two of the corresponding permutation order not to obscure the we. Uniqueness and other properties if two rows are proportional, then the of! The various ways in which objects from a set may be written as function. Construct the corresponding permutation and accessing cookies in your browser has $ 1 $ inversion and so it odd... ), is the same as the parity of the determinant as a... Definition, in contrast to that based on the Laplace expansion, relates to! Of of positive integers not exceeding, with the property that Aw = 0 (. Its determinant, denotedbydet ( a ) v ) that if two rows are proportional, then determinant the. Permutations and the uniqueness of determinants changes that Aw = 0 has property! Introduction to determinant of the corresponding permutation matrix the signum of the determinant gives an N-particle permutations and the of! And accessing cookies in your browser formula from the properties of columns TD 0.0017 Tc [ ( DeÞnition ) (! Order not to obscure the view we leave these proofs for Section 7.3. called its is. Without replacement, to form subsets columns ) of determinants changes of positive not... On three values: 0, 1 ) $ has $ 1 $ inversion and so it odd! Under cookie policy its formal definition ( Leibniz formula ) row-interchanging permutation and uniqueness of determinant matrices, each having determinant....

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