# DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

Save this PDF as:

Size: px
Start display at page:

Download "DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH"

## Transcription

1 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants, lcmda B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices We prove that this quantity is the least common multiple of lcmda) to the power n 1 and certain binomial functions of the entries of A 1 Introduction In a study of non-attacking placements of chess pieces, Chaiken, Hanusa, and Zaslavsky [1] were led to a quasipolynomial formula that depends in part on the least common multiple of the determinants of all square submatrices of a certain Kronecker product matrix, namely, the Kronecker product of an integral 2 2 matrix A with the incidence matrix of a complete graph We give a compact expression for the least common multiple of the subdeterminants of this product matrix, generalized to A of order m 2 2 Background Kronecker product For matrices A = a ij ) m k and B = b ij ) n l, the Kronecker product A B is defined to be the mn kl block matrix a 11 B a 1k B a m1 B a mk B It is known see [2], for example) that when A and B are square matrices of orders m and n, respectively, then deta B) = deta) n detb) m 2000 Mathematics Subject Classification 15A15, 05C50, 15A57 Key words and phrases Kronecker product, determinant, least common multiple, incidence matrix of complete graph, matrix minor 1

2 2 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY The lcmd operation The quantity we want to compute is lcmda B), where for an integer matrix M, the notation lcmdm) denotes the least common multiple of the determinants of all square submatrices of M This is a much stronger question, as the matrices A and B are most likely not square and the result depends on all square submatrices of their Kronecker product We discuss properties of this operation in Section 4, after introducing our main result in Section 3 Incidence matrix For a simple graph G = V,E), the incidence matrix DG) is a V E matrix with a row corresponding to each vertex in V and a column corresponding to each edge in E For a column that corresponds to an edge e = vw, there are exactly two non-zero entries: one +1 and one 1 in the rows corresponding to v and w The sign assignment is arbitrary The complete graph K n is the graph on n vertices v 1,,v n with an edge between every pair of vertices Its incidence matrix has order n n 2) Of interest in this article are Kronecker products of the form A DK n ) Example 1 We present an illustrative example that we will revisit in the proof of our main theorem We consider K 4 to have vertices v 1 through v 4, corresponding to rows 1 through 4 of DK 4 ), and edges e 1 through e 6, corresponding to columns 1 through 6 of DK 4 ) One of the many incidence matrices for K 4 is the 4 6 matrix DK 4 ) = If A is the 3 2 matrix product a 11 a 12 a 21 a 22, we investigate the Kronecker a 31 a 32 A DK 4 ) =

3 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 3 a 11 a 11 a a 12 a 12 a a a 11 a 11 0 a a 12 a a 11 0 a 11 0 a 11 0 a 12 0 a 12 0 a a 11 0 a 11 a a 12 0 a 12 a 12 a 21 a 21 a a 22 a 22 a a a 21 a 21 0 a a 22 a a 21 0 a 21 0 a 21 0 a 22 0 a 22 0 a a 21 0 a 21 a a 22 0 a 22 a 22 a 31 a 31 a a 32 a 32 a a a 31 a 31 0 a a 32 a a 31 0 a 31 0 a 31 0 a 32 0 a 32 0 a a 31 0 a 31 a a 32 0 a 32 a 32 Submatrix notation Let A = a ij ) be an m 2 matrix; this makes A DK n ) an mn nn 1) matrix with non-zero entries ±a ij We introduce new notation for some matrices that will arise naturally in our theorem ) For i,j [m] := {1, 2,,m}, we write A i,j to represent ai1 a i2 If I is a multisubset of [m], we define a a j1 a Ik to be the product j2 i I a ik If I and ) J are multisubsets of [m], we define A I,J to be the ai1 a matrix I2 In this notation, a J1 a J2 lcmda = lcm LCM a ik, LCM deta i,j), i,k i,j where LCM denotes the least common multiple of non-zero quantities taken over all indicated pairs of indices Let 3 Main Theorem and Main Corollary K m := {I,J) : I,J are multisubsets of [m] such that I = J and I J = } Recall that a subdeterminant or minor of a matrix is the determinant of a square submatrix Theorem 2 Let A be an m 2 matrix, not identically zero, and n 1 The least common multiple of all subdeterminants of A DK n ) is lcmd A DK n ) ) [ 1) ] ) = lcm lcmda) n 1, LCM deta I s,j s, K I s,j s ) K

4 4 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY where LCM K denotes the least common multiple of non-zero quantities taken over all collections K K m such that 2 I,J) K I n The proof, which is long, is in Section 7 at the end of this article Although the expression is not as simple as we wanted, we were fortunate to find it; it seems to be a much harder problem to get a similar formula when A has more than two columns Note that it is only necessary to take the LCM component over all maximal collections K, that is, collections K satisfying I s = n/2 When understanding the right-hand side of Equation 1), it may be instructive to notice that the LCM factor on the right-hand side divides deta I,J ) n/2p, disjoint I, J: I = J =p since the largest number of individual deta I,J factors that may occur for disjoint p-member multisubsets I and J of [m] is n/2p When m = 2, the only pair of disjoint p-member multisubsets of [m] is {1 p } and {2 p } From this, we have the following corollary Corollary 3 Let A be a 2 2 matrix, not identically zero, and n 1 The least common multiple of all subdeterminants of A DK n ) is lcmd A DK n ) ) = lcm lcmda) n 1, n/2 LCM p=2 a11 a 22 ) p a 12 a 21 ) p) n/2p ), where LCM denotes the least common multiple over the range of p 4 Properties of the lcmd Operation Four kinds of operation on A do not affect the value of lcmda: permuting rows or columns, duplicating rows or columns, adjoining rows or columns of an identity matrix, and transposition The first two will not change the value of lcmda DK n )) However, the latter two may According to Corollary 3, transposing a 2 2 matrix A does not alter lcmd A DK n ) ) ; but when m > 2 that is no longer the case, as Example 2 shows Adding columns of an identity matrix also may change the lcmd, even when A is 2 2; also see Example 2 However, we may freely adjoin rows of I 2 if A has two columns Corollary 4 Let A be an m 2 matrix, not identically zero, and n 1 Let A be A with any rows of the 2 2 identity matrix adjoined Then lcmd A DK n ) ) = lcmd A DK n ) )

5 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 5 Proof It suffices to consider the case where A is A adjoin an m+1) st row 1 0 ) It is obvious that lcmda = lcmda; this accounts for the first component of the least common multiple in Equation 1) For the second component, any K that appears in the LCM for A also appears for A Suppose K is a collection that appears only for A ; this implies that in K there exist pairs I s,j s ) such that m+1 I s or J s, but that case is similar) Since a Is 2 = 0, deta I s,j s = a Is 1a Jt 2, which is a product of at most n 1 elements of A This, in turn, divides lcmda) n 1 We conclude that the right-hand side of Equation 1) is the same for A as for A We do not know whether or not adjoining a row of the identity matrix to an m l matrix A preserves lcmd A DK n ) ) when l > 2 Limited calculations give the impression that this may indeed be true 5 Examples We calculate a few examples with matrices A that are related to those needed for the chess-piece problem of [1] In that kind of problem the matrix of interest is M DK n ) T where M is an m 2 matrix Hence, in Theorem 2 we want A = M T, so Theorem 2 applies only when m 2 Example 1 When the chess piece is the bishop, M is the 2 2 symmetric matrix ) 1 1 M B = 1 1 We apply Corollary 3 with A = M B, noting that lcmda) = 2 We get lcmd A DK n ) ) = lcm 2 n 1, n/2 LCM p=2 1) p 1 p) n/2p ) The LCM generates powers of 2 no larger than 2 n/2, hence lcmd M B DK n ) ) = 2 n 1 Example 2 When ) the chess piece is the queen, M is the 4 2 I matrix M Q = with lcmda) = 2 Then our matrix A = M M Q T = ) B I MB Since M T Q has four columns Theorem 2 does not apply In fact, we found that lcmd MQ T DK 4) ) = 24, quite different from lcmd M B DK 4 ) ) = 8 However, if we take A = M Q instead of MQ T, Corollary 4 applies; we conclude that lcmd M Q DK n ) ) = lcmd M B DK n ) ) = 2 n 1

6 6 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Thus, A = M Q is an example where transposing A changes the value of lcmd A DK n ) ) dramatically Example 3 A more difficult example is the fairy chess piece known as a nightrider, which moves an unlimited distance in the directions of a knight Here M is the 4 2 matrix 1 2 M N = We can use Theorem 2 to calculate lcmd M N DK n ) Since what is needed for the chess problem is lcmd MN T DK n) ), this example does not help in [1]; nevertheless it makes an interestingly complicated application of Theorem 2 The submatrices ) ) 1 2,, and 1 2 ) 1 2, 2 1 with determinants 3, 4, and 5, respectively, lead to the conclusion that lcmda) = 60 Every pair I, J) of disjoint p-member multisubsets of [4] has one of the following seven forms, up to the order of I and J: {1 q }, {2 r, 3 s, 4 t } ), {2 r }, {1 q, 3 s, 4 t } ), {3 s }, {1 q, 2 r, 4 t } ), {4 t }, {1 q, 2 r, 3 s } ), {1 q, 2 r }, {3 s, 4 t } ), {1 q, 3 s }, {2 r, 4 t } ), {1 q, 4 t }, {2 r, 3 s } ), where the sum of the exponents in each multisubset is p, and where q, r, s, and t may be zero It turns out that det A I,J has the same form in all seven cases: precisely ±2 u 2 2p 2u ±1), where u is a number between 0 and p Furthermore, every value of u from 0 to p appears and every choice of plus or minus sign appears except when u = p) in deta I,J for some choice of I, J) We present two representative examples that support this assertion The case of I = {1 q } and J = {2 r, 3 s, 4 t } Then ) 1 A I,J q 2 = q 2 r 1 s 2 t 1 r 2) s 1) t, with q = r + s + t = p We can rewrite deta I,J as ± 2 s 2 2p s = 2 s 2 2p 2s ± 1) The only instance in where there is no choice of sign is when s = p and r = t = 0, in which case det A I,J simplifies to either 0 or 2 p+1

7 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 7 The case of I = {1 q, 2 r } and J = {3 s, 4 t } Then ) 1 A I,J = q 2 r 2 q 1 r 1 s 2 t 2) s 1) t, where q + r = s + t = p For this choice of I and J, deta I,J = 1) p 2 r+s 2 2p r s Since every deta I,J has the same form, and at most p/2n factors of type 2 2p 2u ±1) may occur at the same time, the LCM in Equation 1) is exactly LCM K I s,j s ) K deta I s,j s ) = 2 N LCM for some N n We conclude that 1 p n/2 0 u p 1 lcmd A DK n ) ) = lcm60 n 1, LCM 1 p n/2 0 u p 1 2 2p 2u ± 1) n/2p, 2 2p 2u ± 1) n/2p ) As a sample of the type of answer we get, when n = 8 this expression is lcmd A DK 8 ) ) = lcm60 7, 4 ± 1) 8/2, 16 ± 1) 8/4, 64 ± 1) 8/6, 256 ± 1) 8/8 ) = The first few values of n give the following numbers: n lcmd A DK 8 ) ) factored) Remarks We hope to determine in the future whether lcmda B) has a simple form for arbitrary matrices A and B Our limited experimental data suggests this may be difficult However, we think at least some generalization of Theorem 2 is possible We would like to understand, at minimum, why the theorem as stated fails when B = DK n ) and A has more than two columns

8 8 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Another direction worth investigating is the number-theoretic aspects of Theorem 2 7 Proof of the Main Theorem During the proof we refer from time to time to Example 1, which will give a concrete illustration of the many steps We assume a 11, a 12, a 21, a 22, a 31, and a 32 are non-zero constants 71 Calculating the determinant of a submatrix Consider an l l submatrix N of A DK n ) We wish to evaluate the determinant of N and show that it divides the right-hand side of Equation 1) We need consider only matrices N whose determinant is not zero, since a matrix with detn = 0 has no effect on the least common multiple Since DK n ) is constructed from a graph, we will analyze N from a graphic perspective The matrix N is a choice of l rows and l columns from A DK n ) This corresponds to a choice of l vertices and l edges from K n where we are allowed to choose up to m copies of each vertex and up to two copies of an edge Another way to say this is that we are choosing m subsets of V K n ), say V 1 through V m, and two subsets of EK n ), say E 1 and E 2, with the property that m i=1 V i = 2 k=1 E k = l From this point of view, if a row in N is taken from the first n rows of A DK n ), we are placing the corresponding vertex of V K n ) in V 1, and so on, up through a row from the last n rows of A DK n ), which corresponds to a vertex in V m We will say that the copy of v in V i is the i th copy of v and the copy of e in E k is the k th copy of e The order of N satisfies l 2n 2 because, if N had 2n 1 columns of A DK n ), then at least one edge set, E 1 or E 2, would contain n edges from K n The columns corresponding to these edges would form a dependent set of columns in N, making detn = 0 In our illustrative example, choose the submatrix N consisting of rows 1, 5, 7, and 10 and columns 1, 4, 7, and 8 Then N is the matrix a 11 0 a 12 a 12 N = a 21 0 a 22 a 22 0 a 21 0 a 22, a 31 a 31 a 32 0 and in the notation above, V 1 = {v 1 }, V 2 = {v 1,v 3 }, V 3 = {v 2 }, E 1 = {e 1,e 4 }, and E 2 = {e 1,e 2 } Returning to the proof, within this framework we will now perform elementary matrix operations on N in order to make its determinant easier to calculate We call the resulting matrix the simplified matrix

9 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 9 of N Each copy of a vertex v has a row in N associated with it; two rows corresponding to two copies of the same vertex contain the same entries except for the different multipliers a ik For example, if v is a vertex in both V 1 and V 2, then there is a row corresponding to the first copy with multipliers a 11 and a 12 and a row corresponding to the second copy with the same entries multiplied by a 21 and a 22 There cannot be a vertex in three or more vertex sets since then the corresponding rows of N would be linearly dependent and det N would be zero When there is a vertex in exactly two vertex sets V i and V j corresponding to two rows R i and R j in N, we perform the following operations depending on the multipliers a i1, a i2, a j1, and a j2 We first notice that deta i,j = a i1 a j2 a i2 a j1 is non-zero; otherwise, the rows R i and R j would be linearly dependent in N and detn = 0 Therefore either both a i1 and a j2 or both a i2 and a j1 are non-zero In the former case, let us add a j1 /a i1 times R i to R j in order to zero out the entries corresponding to edges in E 1 The multipliers of entries in R j corresponding to edges in E 2 are now all deta i,j /a i1 Similarly, we can zero out the entries in R i corresponding to edges in E 2 Lastly, factor out deta i,j /a j2 a i1 from R j If on the other hand, either multiplier a i1 or a j2 is zero, then reverse the roles of i and j in the preceding argument These manipulations ensure that the multiplier of every non-zero entry in N that corresponds to an i th vertex and a k th edge is a ik The appearance of a denominator, a i1 a j2, in the factor deta i,j /a j2 a i1 is merely an artifact of the construction; we could have cancelled it by factoring out a i1 in row i and a j2 in row j However, if we had done this, the entries of the matrix would no longer be of the form a ik, a ik, and 0, which would make the record-keeping in the coming arguments more tedious In our illustrative example, because v 1 is a member of both V 1 and V 2, we perform row operations on the rows of N corresponding to v 1 to yield the simplified matrix of N: a N simplified = 0 0 a 22 a 22 0 a 21 0 a 22 a 31 a 31 a 32 0 The determinants of N and N simplified are related by detn = deta1,2 a 11 a 22 detn simplified

10 10 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY The denominator a 11 a 22 would disappear if we had chosen to factor out the a 11 in the first row and the a 22 in the second row of N simplified Returning to the proof, we assert that the simplified matrix of N has no more that two non-zero entries in any column For a column e corresponding to an edge e = vw in K n, each of v and w is either in one vertex set V i or in two vertex sets V i and V j If the vertex corresponds to two rows in N, the above manipulations ensure that there is only one copy of the vertex that has a non-zero multiplier in the column Another important quality of this simplification is that if a vertex is in more than one vertex set, then every edge incident with one instance of this repeated vertex is now in the same edge set; more precisely, if v V i V j, then every edge incident with the i th copy of v is in E 1 and every edge incident with the j th copy is in E 2, or vice versa Since we are assuming det N 0, N has at least one non-zero entry in each column or row If a row or column) has exactly one nonzero entry, we can reduce the determinant by expanding in that row or column) This contributes that non-zero entry as a factor in the determinant After reducing repeatedly in this way, we arrive at a matrix where each column has exactly two non-zero entries, and each row has at least two non-zero entries This implies that every row has exactly two non-zero entries as well After interchanging the necessary columns and rows and possibly multiplying columns by 1, the structure of what we will call the reduced matrix of N is a block diagonal matrix where each block B is a weighted incidence matrix of a cycle, such as y z 6 z 1 y z 2 y z 3 y z 4 y z 5 y 6 The determinant of a p p matrix of this type is y 1 y p z 1 z p Therefore, we can write the determinant of N as the product of powers of entries of A, powers of deta i,j, and binomials of this form In our illustrative example, we simplify the determinant of N simplified by expanding in the first row contributing a factor of a 11 ), and we perform row and column operations to find the reduced matrix of N to be N reduced = a 21 0 a 22 a 31 a 32 0, 0 a 22 a 22

11 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 11 v q e q v q+1 C Figure 1 An edge e q = v q v q+1 in the cycle C generated by block B When v q V i, v q+1 V j, and e q E k, the contributions y q and z q to detb are a ik and a jk, respectively whose determinant is a 21 a 32 a 22 a 31 a 22 a 22 Returning to the proof, the entries y q and z q are the variables a ik, depending on in which vertex sets the rows lie and in which edge sets the columns lie If the vertices of K n corresponding to the rows in B are labeled v 1 through v p, this block of the block matrix corresponds to traversing the closed walk C = v 1 v 2 v p v 1 in K n in this direction) As a result of the form of the simplified matrix of N, for a column that corresponds to an edge e q = v q v q+1 in E k traversed from the vertex v q in vertex set V i to the vertex v q+1 in vertex set V j, the entry y q is a ik and the entry z q is a jk See Figure 1) Therefore each block B in the block diagonal matrix contributes 2) detb = a ik e=v q v q+1 C e E k,v q V i a jk e=v q v q+1 C e E k,v q+1 V j for some closed walk C in G, whose length is p In our illustrative example, N reduced is the incidence matrix of the closed walk v 3 e 4 v2 e 1 v1 e 2 v3, where vertex v 3 is from V 2, vertex v 2 is from V 3, and vertex v 3 is from V 2 Moreover, edge e 4 is from E 1, and edges e 1 and e 2 are from E 2 Because we are working with a concrete example, we have not relabeled the vertices as we did in the preceding paragraph Returning to the proof, we can simplify this expression by analyzing exactly what the a ik and a jk are Suppose that two consecutive edges e q 1 and e q in C are in the same edge set E k, and suppose that the vertex v q that these edges share is in V i See Figure 2) In this case, both entries z q 1 and y q are a ik, which can then be factored out of each product in Equation 2) A particular case to mention is when the cycle C contains a vertex that has multiple copies in N not necessarily both in C) In this case,

12 12 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY v q e q 1 e q C Figure 2 Two consecutive edges e q 1 and e q, both incident with vertex v q in the cycle C generated by block B When both edges are members of the same edge set E k and v q is a member of V i, the contributions z q 1 and y q are both a ik, allowing this multiplier to be factored out of Equation 2) the edges of C incident with this repeated vertex are both from the same edge set, as mentioned earlier After factoring out a multiplier for each pair of adjacent edges in the same edge set, all that remains inside the products in Equation 2) is the contributions of multipliers from vertices where the incident edges are from different edge sets More precisely, when following the closed walk, let I be the multiset of indices i such that the walk C passes from an edge in E 2 to an edge in E 1 at a vertex in V i Similarly, let J be the multiset of indices j such that C passes from an edge in E 1 to an edge in E 2 at a vertex in V j Then what remains inside the products in Equation 2) after factoring out common multipliers is exactly deta I,J = a i1 a j2 a j2 a i2 i I j J j J There is one final simplifying step Consider a value i occurring in both I and J In this case, we can factor a i1 a i2 out of both terms This implies that the determinant of each block B of the block diagonal matrix is of the form 3) ± i,k a s ik ik ) deta I,J, where the exponents s i,k are non-negative integers, I and J are disjoint subsets of [m] of the same cardinality, and 2 I + i,k s ik = p because the degree of detb is the order of B Notice that when I = J = 1 say I = {i} and J = {j}), the factor deta I,J equals deta i,j Combining contributions from the simplification and reduction processes and from all blocks, we have the formula 4) detn = ± i,j deta i,j ) V i V j i,k a S ik ik i I deta I B,J B, B

13 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 13 for some non-negative exponents S ik We note that 2 Vi V j + S ik + I B = l In the cycle in our illustrative example, vertex v 3 in V 2 ) transitions from an edge in E 2 to an edge in E 1 and vertex v 2 in V 3 ) transitions from an edge in E 1 to an edge in E 2 This implies that I = {2} and J = {3} Vertex v 1 originally occurred in the two vertex sets V 1 and V 2 ; this implies that we can factor out the corresponding multiplier, a 22 Indeed, the determinant of N reduced is a 21 a 32 a 22 a 2 22a 31 = a 22 deta 2,3 Through these calculations we see that ) deta 1,2 detn = a 11 a 22 deta 2,3 = det A 1,2 deta 2,3 a 11 a The subdeterminant divides the formula We now verify that the product in Equation 4) divides the right-hand side of Equation 1) The exponents S ik can be no larger than n because there are only n rows with entries a ik in A DK n ), so the expansion of the determinant, as a polynomial in the variable a ik, has degree at most n Furthermore, it is not possible for the exponent of a ik to be n The only way this might occur is if N were to contain in V i all n vertices of G and at least n edges of E k incident with the vertices of V i The corresponding set of columns is a dependent set of columns in N because the rank of DK n ) is n 1), which would make detn = 0 Therefore, detn contributes no more than n 1 factors of any a ik to any term of lcmda DK n )) Now let us examine the exponents of factors of the form det A I,J that may divide detn Such factors may arise either upon the conversion of N to the simplified matrix of N if I = J = 1, or from a block of the reduced matrix as in Equation 3) if I = J 1 The factors that arise in simplification come from duplicate pairs of vertices: every duplicated vertex v leads to a factor deta i,j where v V i V j this is apparent in Equation 4)) The total number of factors deta i,j arising from simplification is {i,j} V i V j = d, the number of duplicated vertices, which is not more than n 1 since 2d l 2n 1) Since each such factor divides lcmda, their product divides lcmda) n 1, the first component of Equation 1) The factors deta I B,J B from blocks B of the reduced matrix arise from simple vertices those which are not duplicated among the rows of N I B and J B are multisets of indices of vertex sets V i containing simple vertices, I B J B =, and B I B + J B ) c, the number of simple vertices, since a simple vertex appears in only one block As

14 14 CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY c n, B I B + J B ) n Thus, the product of the corresponding determinants deta I B,J B is one product in the LCM K component of Equation 1) 73 The formula is best possible We have shown that for every matrix N, detn divides the right-hand side of Equation 1) We now show that there exist graphs that attain the claimed powers of factors Consider the path of length n 1, P = v 1 v 2 v n, as a subgraph of K n Create the 2n 2) 2n 2) submatrix N of A DK n ) with rows corresponding to both an i th copy and a j th copy of vertices v 1 through v n 1 and columns corresponding to two copies of every edge in P Then a i a i a i1 a i1 0 0 a i2 a i a N = i1 a i1 0 0 a i2 a i2, a j a j a j1 a j1 0 0 a j2 a j a j1 a j1 0 0 a j2 a j2 with determinant deta i,j ) n 1 The four quadrants of N are n 1) n 1) submatrices of A DK n ) with determinants a n 1 i1, a n 1 i2, a n 1 j1, and a n 1 j2, respectively We show that, for every collection K = {I s,j s )}) K m satisfying 2 I s n, there is a submatrix N of A DK n ) with determinant I s,j s ) K detai s,j s For each s, starting with s = 1, choose W s V K n ) to consist of the lowest-numbered unused n s = 2 I s vertices Thus, W s = {v 2k+1,,v 2k+2ns } Take edges v i 1 v i for 2k + 1 < i 2k + 2n s and v 2k+1 v 2k+2ns This creates a cycle C s if I s > 1 and an edge e s if I s = 1 For a cycle C s, place each odd-indexed vertex of W s into a vertex set V i for every i I s and each even-indexed vertex into a vertex set V j for every j J s For an edge e s corresponding to I s = {i} and J s = {j}, place both vertices of W s in V i and V j Place all edges of the form v 2m 1 v 2m into E 1 and all edges of the form v 2m v 2m+1 and v 2k+1 v 2k+2ns into E 2 Note that this puts e s into both E 1 and E 2 The submatrix N of A DK n ) that arises from placing the vertices in numerical order and the edges in cyclic order along C s is the blockdiagonal matrix where each block N s is a 2 I s 2 I s matrix of the

15 DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES 15 form a i a i1 2 a j1 1 a j a i2 2 a i a j2 1 a j a jl 1 a jl 2 if C s is a cycle and ) ai1 a i2 a j1 a j2 if e s is an edge The determinant of N s is exactly deta I s,j s, so the determinant of N is I s,j s ) K detai s,j s, as desired 8 Acknowledgements The authors would like to thank an anonymous referee whose suggestions greatly improved the readability of our article References [1] Seth Chaiken, Christopher RH Hanusa, and Thomas Zaslavsky, A q-queens problem In preparation [2] Roger A Horn and Charles R Johnson Topics in Matrix Analysis Cambridge University Press, New York 1991 vii pp Department of Mathematics, Queens College CUNY), Kissena Blvd, Flushing, NY 11367, USA, phone: address: Department of Mathematical Sciences, Binghamton University SUNY), Binghamton, NY , USA address:

### Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

FPSAC 2009 DMTCS proc (subm), by the authors, 1 10 Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph Christopher R H Hanusa 1 and Thomas Zaslavsky 2 1 Department

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### Unit 18 Determinants

Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of

### 2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### Determinants. Chapter Properties of the Determinant

Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

### 5.3 Determinants and Cramer s Rule

290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Sign pattern matrices that admit M, N, P or inverse M matrices

Sign pattern matrices that admit M, N, P or inverse M matrices C Mendes Araújo, Juan R Torregrosa CMAT - Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### 1 Determinants. Definition 1

Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### THE SIGN OF A PERMUTATION

THE SIGN OF A PERMUTATION KEITH CONRAD 1. Introduction Throughout this discussion, n 2. Any cycle in S n is a product of transpositions: the identity (1) is (12)(12), and a k-cycle with k 2 can be written

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### 1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Chapter 5. Matrices. 5.1 Inverses, Part 1

Chapter 5 Matrices The classification result at the end of the previous chapter says that any finite-dimensional vector space looks like a space of column vectors. In the next couple of chapters we re

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### Suk-Geun Hwang and Jin-Woo Park

Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGN-SOLVABILITY Suk-Geun Hwang and Jin-Woo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### *.I Zolotareff s Proof of Quadratic Reciprocity

*.I. ZOLOTAREFF S PROOF OF QUADRATIC RECIPROCITY 1 *.I Zolotareff s Proof of Quadratic Reciprocity This proof requires a fair amount of preparations on permutations and their signs. Most of the material

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

### 1 Scalars, Vectors and Tensors

DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

### INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

MODULAR ARITHMETIC KEITH CONRAD. Introduction We will define the notion of congruent integers (with respect to a modulus) and develop some basic ideas of modular arithmetic. Applications of modular arithmetic

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary