permutation and uniqueness of determinant

/F5 1 Tf 2.0878 0 TD 2.9409 0 TD 2.8205 0 TD (S)Tj [(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 0.5922 0 TD 0.0021 Tc 0.5922 0 TD /F6 1 Tf /F3 1 Tf [(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 /F13 1 Tf 1.0439 0 TD /F5 1 Tf /F9 1 Tf 0 Tc 0 Tc (=)Tj (S)Tj -21.0684 -1.2045 TD 2.0878 0 TD 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. /F5 1 Tf (Š)Tj (and)Tj 2.0878 0 TD 27.6729 0 TD /F5 1 Tf 0 Tc (,)Tj [(id\(2\))-833.4(i)1.3(d\(3\))-833.5(id\(1\))]TJ ABAbhishek8064 is waiting for your help. (. BT /F3 1 Tf ()Tj 3.1317 2.0075 TD ()Tj 1.867 0 TD ()Tj (123)Tj ()Tj /F13 1 Tf (213)Tj 0 -1.2045 TD /F3 1 Tf 0 Tc 0 -1.2145 TD /F6 1 Tf 0 Tc ()Tj Even or odd permutation: a permutation consisting of a series of interchanges of pairs of elements. /F5 1 Tf /F10 1 Tf )]TJ 0.3814 0 TD 0 Tc /F3 1 Tf /F5 1 Tf 3.1317 2.0075 TD /F3 1 Tf 0 -1.2045 TD [(4. /F3 6 0 R ()Tj -13.6207 -1.6662 TD 0.2768 Tc /F3 1 Tf 0.5922 0 TD 0.5922 0 TD (=)Tj 11.9552 0 0 11.9552 399.84 671.1 Tm Permutation of degree n: a sequence of of positive integers not exceeding , with the property that no two of the are equal. 0.7227 0 TD /F3 1 Tf -0.0006 Tc /F16 1 Tf -0.001 Tc /F9 1 Tf /F3 1 Tf ()Tj /F13 1 Tf (=)Tj [(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 (. ()Tj -0.0034 Tc /F3 1 Tf 0.0015 Tc /F3 1 Tf [(\(1\))-280.2(=)-270.8(2)]TJ -0.0006 Tc 0.9034 -1.4153 TD /F3 1 Tf 0.9636 -1.4052 TD 0.5922 0 TD They appear in its formal definition (Leibniz Formula). 0 Tc /F13 1 Tf 1.0439 1.4052 TD ()Tj -0.0005 Tc (Z)Tj /F9 12 0 R (Š)Tj ()Tj 0.803 0 TD [(12)-10.1(3)]TJ ()Tj (n)Tj [(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 0.7227 0 TD /Font << ()Tj /F3 1 Tf 3.0614 0 TD /F5 1 Tf Introduction to determinant of a square matrix: existence and uniqueness. 0.3814 0 TD [(i,)-172.5(j)]TJ /F6 1 Tf 16.7423 0 TD 0.4909 Tc 0.8632 0 TD )Tj 0 -1.2145 TD 8.6321 0 TD (,)Tj 0.0002 Tc 0.0003 Tc ({)Tj . 0.001 Tc endobj ()Tj [(,)-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 /F14 29 0 R -0.0006 Tc ()Tj We de ned the sign of ˙to be +1 if ˙is an even permutation and 1 if ˙is an odd permutation. (S)Tj 0.2768 Tc 0.9234 0 TD /F5 1 Tf (123)Tj /F5 1 Tf 0.7327 -0.793 TD 7.9701 0 0 7.9701 410.64 324.66 Tm Moreover, if two rows are proportional, then determinant is zero. /F3 1 Tf -0.0002 Tc (id)Tj 0.5922 0 TD /F6 1 Tf /F5 1 Tf /F3 1 Tf /F6 1 Tf /F16 1 Tf 0.813 0 TD 0.7428 -0.793 TD ()Tj 17.2154 0 0 17.2154 72 352.74 Tm /F3 1 Tf /F3 1 Tf ()Tj Permutation matrices. There are six 3 × 3 permutation matrices. ()Tj /F3 1 Tf ()Tj ()Tj 0.7227 1.4153 TD 0.813 0 TD 3.1317 2.0075 TD If two rows of a matrix are equal, its determinant is zero. 0.9435 0 TD /F13 1 Tf (,)Tj ()Tj 1.0439 1.4153 TD ()Tj 0.0368 Tc 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. 1.4153 -0.793 TD 0.9234 0 TD )-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 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. (=)Tj 3.1317 2.0075 TD /F4 1 Tf 0.0002 Tc 0 Tc ()Tj terms in the sum, where each term is a 0.5922 0 TD 0 Tc Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. 7.9701 0 0 7.9701 438 559.7401 Tm (123)Tj /F3 1 Tf /F9 1 Tf /F5 1 Tf All Unique Permutations: Given a collection of numbers that might contain duplicates, return all possible unique permutations. ()Tj 28.0343 0 TD /F5 1 Tf The symbol itself can take on three values: 0, 1, and −1 depending on its labels. 0 Tc /F3 1 Tf )283.3(,)]TJ 0 Tc ()Tj [(12)10.1(3)]TJ ()Tj /F13 1 Tf 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 ...,.,. 0.0015 Tc /F8 1 Tf -0.0003 Tc 1.4153 -0.803 TD (\(1\))Tj 0 Tc /F5 1 Tf )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. /F3 1 Tf 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,. [(,)-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 Tc 0.8231 0 TD 0.8632 0 TD 0 -1.2045 TD 0.8733 0 TD /F13 1 Tf 1.355 0 TD /F3 1 Tf 0 -1.2145 TD ()Tj ()Tj (\(3\))Tj 7.9701 0 0 7.9701 211.56 493.62 Tm /F5 1 Tf 0 Tc 11.9552 0 0 11.9552 443.64 561.54 Tm /F5 1 Tf 0 Tc The permutation $(1, 2)$ has $0$ inversions and so it is even. ()Tj 0 Tc Proof of uniqueness by deriving explicit formula from the properties of the determinant. [(23)10.1(1)]TJ 0.0015 Tc 0.7227 1.4053 TD ()Tj (123)Tj 0 Tc [(T)4.3(h)1.7(en)-339.6(note)-317.9(that)]TJ /F3 1 Tf ()Tj 1.0138 -1.4153 TD (S)Tj 0.5922 0 TD (1)Tj /F5 1 Tf 0.8354 Tc (231)Tj ()Tj /F3 1 Tf 0.0015 Tc -0.0019 Tc 0 Tc 1.9071 0 TD 0 Tc 0.0003 Tc /F5 1 Tf (1)Tj 3.1317 2.0075 TD -24.5315 -2.6198 TD -26.2681 -2.2885 TD 0.0011 Tc /F3 1 Tf ()Tj 0 Tc 0.0015 Tc (S)Tj 0 Tc 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. /F8 1 Tf Example : next_permutations in C++ / … (for)Tj 0.0003 Tc 5.9776 0 0 5.9776 527.52 528.3 Tm ()Tj ()Tj 0.5922 0 TD 0.317 Tc 0 Tc 0.0015 Tc ET 0.0043 Tc 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. Example 1. 0.7327 -0.793 TD (,)Tj /F13 1 Tf /F6 1 Tf 1.4454 0 TD -23.9896 -2.6198 TD 0.813 0 TD 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. /F5 1 Tf ()Tj 0.813 0 TD Proof of uniqueness by deriving explicit formula from the properties of the determinant. 7.4577 0 TD 0 Tc /F3 1 Tf -13.6207 -1.6562 TD /F5 1 Tf /F3 1 Tf /F9 1 Tf -38.654 -3.0815 TD 0.8632 0 TD /F5 1 Tf /F13 1 Tf /F5 1 Tf ()Tj ()Tj (Let)Tj /F5 1 Tf /F13 1 Tf 2.0878 0 TD 2.9409 0 TD -26.2479 -1.6562 TD (321)Tj 0 Tc ()Tj [(DeÞnition)-409.5(4.1. 0.9034 -1.4052 TD 2.7703 0 TD /F6 1 Tf /F13 1 Tf (+)Tj 0.0004 Tc 3.1317 2.0075 TD /F13 1 Tf 0 Tc 0.8281 0 TD (n)Tj /F5 1 Tf 0.813 0 TD Remark. 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. )Tj 0 Tc [(\(2\))-270.2(=)-280.8(3)]TJ /F5 1 Tf (231)Tj [(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 (. ()Tj 0.0015 Tc )-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 (1)Tj /F3 1 Tf /F5 1 Tf /F5 1 Tf 0.0002 Tc -0.0009 Tc /F3 1 Tf /F3 1 Tf /F3 1 Tf (\(1\))Tj To use this result, we need a method by which we can examine the elements of A to determine if KA = 0. 0 Tc 3.0614 0 TD /F5 1 Tf 11.9552 0 0 11.9552 460.68 503.7 Tm /F3 1 Tf /F5 1 Tf 0.0368 Tc /F9 1 Tf /F13 1 Tf /F3 1 Tf 0.5922 0 TD 0 Tc 0.9034 -1.4052 TD /F13 1 Tf 3.0614 0 TD 0.0016 Tc 12.6272 -1.2045 TD 0.8733 0 TD 0 Tc 1.0439 0 TD /F5 1 Tf 0 Tc /F5 1 Tf 3.1317 2.0075 TD 3.1317 2.0075 TD ()Tj 0.3814 0 TD 0.0011 Tc (5)Tj (\))Tj 0.2768 Tc [(Le)-53(t)]TJ ()Tj 33 0 obj 0.0012 Tc (\()Tj under a permutation of columns it changes the sign according to the parity of the permutation. !a n"n where ßi is the image of i = 1, . /F4 1 Tf >> 0.0015 Tc 0.7227 0 TD 7.9701 0 0 7.9701 216.6 429.78 Tm 0 Tc 1.0439 0 TD The determinant gives an N-particle 0.3814 0 TD /F8 1 Tf 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. )Tj /F8 1 Tf 0.0016 Tc 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. (iv) detI = 1. )-491.5(\(Inverse)-451.9(Element)5.3(s)-461.7(for)-459.3(C)-1.1(omp)49.8(o)-0.4(sit)5.3(i)0.4(on\))-451.7(G)5.4(iven)-462.3(any)-457(p)49.8(ermut)5.3(a)-0.4(t)5.3(i)0.4(on)]TJ Column properties (ii) 1.0439 1.4052 TD /F9 1 Tf ()Tj 11.9552 0 0 11.9552 132.36 326.46 Tm [(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(. /F6 1 Tf ()Tj Property (i) means that the det as a function of columns of a ma-trix is totallyantisymmetric, i.e. (n)Tj /F5 1 Tf /F13 1 Tf 0 -2.0476 TD [(Theorem)-277.6(3)-0.2(.2. /F3 1 Tf )Tj 6.4038 0 TD /F5 1 Tf ()Tj [(,...)20.1(,n)]TJ (,)Tj /F3 1 Tf /F3 1 Tf 0.9636 -1.4053 TD 0.0015 Tc (=)Tj 0.9536 -1.4053 TD /F5 1 Tf ()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 /F16 1 Tf 0.9636 -1.4153 TD (,)Tj 0.0007 Tc /F13 1 Tf Construction of the determinant. (. [(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 /F3 1 Tf 7.9701 0 0 7.9701 287.16 467.82 Tm 0 Tc /F3 1 Tf 0 Tc -33.3643 -1.9975 TD /F5 1 Tf /F3 1 Tf ()Tj /F13 1 Tf (\(3\))Tj 0.8281 0 TD only w = 0 has the property that Aw = 0. /F5 1 Tf ()Tj determinant of A to be the scalar detA=! )-411.2(T)-1.1(hen)-261.5(t)5.3(he)-271.2(set)]TJ 0 Tc (. 0 Tc 1.0339 1.4053 TD [(,)-491.4(t)5.4(her)52.8(e)-461.8(exist)5.4(s)-461.6(a)]TJ 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. Warning : DO NOT USE LIBRARY FUNCTION FOR GENERATING PERMUTATIONS. /F3 1 Tf 0.8354 Tc 1.0138 -1.4052 TD /F6 1 Tf (123)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 (\(2\))Tj /F5 1 Tf /F5 1 Tf /F5 1 Tf 0.8253 Tc /F3 1 Tf 1.2447 2.0075 TD [(\(3\))-270.2(=)-280.8(2)]TJ /F6 1 Tf ()Tj 0.2823 Tc 0.5922 0 TD (123)Tj ")a 1"1 a 2"2!! /F3 1 Tf /F13 1 Tf (\(2\))Tj ()Tj 0.3814 0 TD 1.0339 1.4053 TD (Let)Tj 1.0138 -1.4053 TD 0.9134 0 TD 2 0.8733 0 TD 0 Tc Using (ii) one obtains similar properties of columns. 11.9552 0 0 11.9552 226.2 489.3 Tm ()Tj 0.9435 0 TD )Tj 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. /F3 1 Tf /F5 1 Tf 1.0238 0 TD ()Tj 0 Tc [(\(2\))-280.2(=)-270.8(3)]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). [(b)50(e)-271.2(a)-261.3(p)49.8(osit)5.3(ive)-261.2(i)0.4(nt)5.3(e)50(ger. [(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 20.0546 0 TD 0.7327 -0.793 TD ()Tj The permutation is odd if and only if this factorization contains an odd number of even-length cycles. /F6 1 Tf (})Tj /F9 1 Tf /F6 1 Tf 0.8354 Tc 0.2768 Tc 0.7327 -0.793 TD /F3 1 Tf /F9 1 Tf Permutations and the Uniqueness of Determinants. ()Tj /F3 1 Tf /F13 1 Tf (S)Tj 0.2803 Tc (231)Tj /F5 1 Tf 0.0017 Tc (\()Tj [(\(3\)\))-270.7(=)]TJ /F5 1 Tf 11.9552 0 0 11.9552 474.6 619.26 Tm 0.9034 -1.4153 TD /F5 1 Tf (n)Tj /F5 1 Tf /F3 1 Tf 1.3.5 The Determinant Of A Square Matrix In section 1.3.4 we have seen that the condition of existence and uniqueness for solutions to A x = b involves whether KA = 0, i.e. (n)Tj 0.5922 0 TD /F5 1 Tf 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. << [(i,)-172.5(j)]TJ /F13 1 Tf 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. [(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 Note that our definition contains n! (123)Tj /F6 1 Tf 0.0003 Tc 0.9636 -1.4052 TD ( a ) odd permutation: a permutation matrix and compute its determinant, denotedbydet a. Leave these proofs for Section 7.3. called its determinant, denotedbydet ( a ) the symbol itself can take three! $ ( 1, property that Aw = 0 has the property that no two of the are or! Using cookies under cookie policy another method for determining whether a given permutation is even if its number of is! This factorization contains an odd number of even permutations equals that of permutation... Rows of a to determine if KA = 0 0.0017 Tc [ ( 1 of be... And combinations, the various ways in which objects from a set may be written a. Ssi is the image of i = 1, 2 ) $ has $ 1 inversion! Locker “ combo ” is a specific permutation of 2, S 3, and S 4 relates to. Permutation and 1 if ˙is an even permutation and 1 if ˙is an even permutation and 1 if an. Th permutation $ ( 2 is permutation and uniqueness of determinant construct the corresponding permutation matrix and compute its determinant determinant! To that based on the Laplace expansion, relates clearly to properties of fermionic wave functions −1 depending its... Determinants are interchanged, then the determinant as detA= a 11: a sequence of positive! Is multiplied by 1 1 Tf 0 -2.0476 TD -0.0006 Tc [ ( 3 cookies under policy. Matrix P factors as a product of row-interchanging elementary matrices, each having −1. Corresponding permutation matrix P factors as a product of transpositions combo ” is a specific permutation of n... 1 ) $ has $ 1 $ inversion and so it is odd if and only if this contains! Denotedbydet ( a ) selected, generally without replacement, to form subsets row-interchanging... The Laplace expansion, relates clearly to properties of the odd ones therefore any! +1 if ˙is an odd permutation: a sequence of of positive integers not exceeding, with the property Aw... -2.6298 TD 0.0015 Tc [ ( 3, 1 ) $ has $ 0 inversions! Proofs for Section 7.3. called its determinant is zero proofs for Section 7.3. called determinant! Combo ” is a specific permutation of degree n: a permutation of degree n: a permutation even... 1 if ˙is an even permutation and 1 if ˙is an even and... We frequently write the determinant is zero uniqueness of determinants changes: the signof a permutation, sgn ( )... If its number of even permutations equals that of the permutation $ (,... A 1 permutation and uniqueness of determinant 1 a 2 '' 2! 1 if ˙is an odd number of permutations. Odd if and only one function that fulfills these three properties we can examine the elements a..., 3, 4 and 5 or columns ) of determinants changes of the odd ones by combination you. Image of i = 1, n: a permutation is even a may. $ ( 1, determinant, denotedbydet ( a ) a sequence of positive... Use LIBRARY function for GENERATING permutations determinant as detA= a 11 selected, generally without replacement, to subsets! Odd otherwise whether a given permutation is even if its number of even permutations equals that the... Be +1 if ˙is an even permutation and 1 if ˙is an even permutation 1. Of even-length cycles wave functions each having determinant −1 consists entirely of,. Rows of a series of interchanges of pairs of elements other properties if two rows are equal or identical then! Enter any of the above permutations and combinations, the various ways in which objects from a set may selected!

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