A Remark on the Manhattan Distance Matrix of a Rectangular Grid

Home , Permutation matrix

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1.1 Notation The positive semideﬁniteness of a symmetric real matrix A is denoted by A 0. The set of symmetric real matrices of order n is denoted by n. In denotes the identity S n matrix of order n and en denotes the vector of all 1’s in R . En denotes the n n T × matrix of all 1’s, i.e., En = enen . The subscript n will be omitted if the dimension of In, en and En is clear from the context. A† denotes the Moore-Penrose inverse of A. For two matrices A and B, A B denotes the Kronecker product of A and B. For a real ⊗ number x, x denote the ceiling of x. Finally, diag(A) denotes the vector consisting ⌈ ⌉ of the diagonal entries of a matrix A.

2 Preliminaries

The following well-known [3] lemmas will be needed in some of the proofs below.

Lemma 2.1 Let A and B be two n n real matrices. × T T T 1. (A )†= (A†) := A† .

T T 2. (AA )†= A† A†.

T T 3. If A B = 0 and BA = 0, then (A + B)† = A† + B†. 4. If AT B = 0 and BAT = 0, then rank (A + B) = rank A + rank B.

T 1 T 5. If A has full column rank then A† = (A A)− A . 6. Let C and Q be two real matrices of orders k k and n k respectively, where T T T × × Q Q = Ik. Then (QCQ )† = QC†Q .

Lemma 2.2 Let A and B be two n n real matrices. × 1. (A B) = A B . ⊗ † † ⊗ † 2. Let A 0 and B 0. Then (A B) 0. ⊗

2 3. rank(A B) = rank (A) rank (B). ⊗ Lemma 2.3 (Generalized Schur Complement) Given the real symmetric parti- tioned matrix A B M = . BT C Then M 0 if and only if C 0, A BC BT 0, and the null space of C is a subset − † of the null space of B.

3 Euclidean Distance Matrices (EDMs)

An n n matrix D = (dij ) is called a Euclidean distance matrix (EDM) if there exist × 1 points p ,...,pn in some Euclidean space Rr such that

2 d = pi pj for all i, j = 1,...,n, ij || − || where denotes the Euclidean norm. Moreover, the dimension of the aﬃne span of || || 1 the points p ,...,pn is called the embedding dimension of D. Without loss of generality, we make the following assumption.

Assumption 3.1 The origin coincides with the centroid of the points p1,...,pn.

It is well known [12, 14] that a symmetric matrix D whose diagonal entries are all 0’s is an EDM if and only if D is negative semideﬁnite on the subspace

M := x Rn : eT x = 0 , { ∈ } where e is the vector of all 1’s in Rn. It easily follows that the orthogonal projection on M is given by J := I eeT /n = I E/n. (1) − − Let denote the set of symmetric n n real matrices and let and be two Sn × SH SC subspaces of such that: Sn = A : diag(A)= 0 and = A : Ae = 0 . SH { ∈ Sn } SC { ∈ Sn } Furthermore, let [5] : and : be the two linear maps deﬁned by T SH → SC K SC → SH

1 (D) := JDJ, (2) T −2 (B) := diag(B) eT + e (diag(B))T 2 B. (3) K −

3 Then it immediately follows that and are mutually inverse between the two T K subspaces and ; and that D is an EDM of embedding dimension r if and only if SH SC the matrix (D) is positive semidefinite of rank r [5]. Moreover, it is not difficult to T show (see, e.g., [1, 6, 13]) that if D is an EDM of embedding dimension r then rank (D) is equal to either r +1 or r + 2. Note that if D is an EDM of embedding dimension r then (D) is the Gram matrix T of the points p1,...,pn, i.e., (D)= P P T , where P is n r and piT is the ith row of T × P . i.e., p1T  .  P = . . (4)  pnT    P is called a configuration matrix of D. Also, note that P T e = 0 which is consistent with our assumption that the origin coincides with the centroid of the points p1,...,pn.

3.1 Spherical EDMs An EDM D is called spherical if the points p1,p2,...,pn that generate D lie on a hypersphere. Otherwise, D is called non-spherical. The following characterization of spherical EDMs is well known.

Theorem 3.1 ([10, 6, 13]) Let D = 0 be a given n n EDM with embedding dimen- 6 × sion r, and let P be a configuration matrix of D such that P T e = 0. Then the following statements are equivalent. 1. D is a spherical EDM. 2. Rank (D) = r + 1. 3. The matrix λ eeT D is positive semidefinite for some scalar λ. − 4. There exists a vector a in Rr such that: 1 P a = Jdiag ( (D)), (5) 2 T where J is as defined in (1).

Two remarks are in order here. First, spherical EDMs can also be characterized in terms of Gale transform [2] which lies outside the scope of this paper. Second, if D is an n n spherical EDM then the points that generate D lie on a hypersphere of center × 2 1 2 a and radius ρ = (aT a + eT De/2n ) / , where a is given in (5) (see [13]). Furthermore, as the next theorem shows, the minimum value of λ such that λE D 0 is closely − related to ρ.

4 Theorem 3.2 ([10, 13]) Let D be a spherical EDM generated by points that lie on a hypersphere of radius ρ. Then λ = 2ρ2 is the minimum value of λ such that λE D ∗ − 0.

The following lemma gives an explicit expression for the radius ρ which is more conve- nient for our purposes.

Lemma 3.1 Let D be a spherical EDM generated by points that lie on a hypersphere of radius ρ. Then T T 2 e De e D( (D)) De ρ = + T † . (6) 2n2 4n2 e Proof. Let D be a spherical EDM and let Q = [ √n V ] be an orthogonal matrix, i.e., J = VV T . Then λE D 0 if and only if QT (λE D)Q 0. But − − λn eT De/n eT DV/√n QT (λE D)Q = − − . − V T De/√n V T DV − − Clearly V T DV = 2V T (D)V 0. Thus it follows from Lemma 2.3 that λE D 0 − T − if and only if λn eT De/n eT DV (V T (D)V ) V T De/2n 0 and the null space of − − T † ≥ V T DV is a subset of the null space eT DV . But it follows from Property (6) of Lemma 2.1 that V (V T (D)V ) V T = (VV T (D)VV T ) = (D) since VV T = J and since T † T † T † ( (D))e = 0. Moreover, since D is a spherical EDM, it follows that the null space T of V T DV is a subset of the null space eT DV [2]. Thus, λE D 0 if and only if T T − λn e De/n e D( (D))†De/2n 0 and the result follows by Theorem 3.2. − − T ≥ ✷ T T 2 T Note that ( (D))†=(P P )† = P † P †. Thus it follows from (6) that ρ = a a + T 2 TT 2 T T 2 e De/2n = e De/2n + e DP † P †De/4n . Therefore,

1 1 T 1 T a = P †De = (P P )− P De, (7) 2n 2n where P is a conﬁguration matrix of D. An interesting class of spherical EDMs is that of regular EDMs. A spherical EDM D is said to be regular (also called EDM of strength 1 [9, 10]) if D is generated by points that lie on a hypersphere centered at the centroid of these points [7]. i.e., na = P T e = 0, since we assume that the centroid of the points p1,...,pn is located at the origin. Therefore, if D is a regular EDM then diag( (D)) = ρ2e. Hence, it follows from (3) T that 2 De = 2nρ e, where ρ2 = eT De/2n2. Thus we have the following characterization of regular EDMs

5 Theorem 3.3 ([7, 9, 10]) Let D be an EDM. Then D is regular if and only if e is an eigenvector of D.

4 Main Results

Theorem 4.1 Let D1 be an m m EDM of embedding dimension r1, and let D2 be an × n n EDM of embedding dimension r2. Then D = E D2 + D1 E is an EDM of × m ⊗ ⊗ n embedding dimension r1 + r2, where E is the m m matrix of all 1’s and denotes m × ⊗ the Kronecker product.

Proof. I = I I and E = E E . Thus nm m ⊗ n nm m ⊗ n 1 1 1 (E D2) = (I I E E )(E D2)(I I E E ) T m ⊗ −2 m ⊗ n − nm m ⊗ n m ⊗ m ⊗ n − nm m ⊗ n 1 1 1 = E (I E )D2(I E ) −2 m ⊗ n − n n n − n n = E (D2) 0. m ⊗T Similarly (D1 E )= (D1) E 0. Therefore, (D)= E (D2)+ (D1) T ⊗ n T ⊗ n T m ⊗T T ⊗ E 0. Hence D is EDM. n On the other hand, we have from Lemma 2.1 that rank ( (D)) = rank (E T m ⊗ (D2)+ (D1) E ) = rank (E (D2)) + rank ( (D1) E ) = rank ( (D2)) + T T ⊗ n m ⊗T T ⊗ n T rank ( (D1)) = r2 + r1. Thus the embedding dimension of D is equal to r1 + r2. T ✷

Theorem 4.2 Let D1 be a spherical EDM of order m generated by points that lie on a hypersphere of radius ρ1, and let D2 be a spherical EDM of order n generated by points that lie on a hypersphere of radius ρ2. Then D = Em D2 + D1 En is a spherical ⊗ ⊗2 2 1/2 EDM generated by points that lie on a hypersphere of radius ρ = (ρ1 + ρ2) .

Proof. λEnm D = Em (λ2En D2) + (λ1Em D1) En where λ = λ1 + λ2. − ⊗ − − ⊗ 2 Since D1 and D2 are spherical EDMs it follows that λ2En D2 0 for λ2 = 2ρ2, and 2 2 2 − λ1E D1 0 for λ1 = 2ρ1. Therefore, for λ = 2ρ1 + 2ρ2 we have λE D 0 and n − nm − hence D is a spherical EDM generated by points that lie on a hypersphere of radius 2 2 1/2 2 2 1/2 ρ (ρ1 + ρ2) . Next we show that ρ = (ρ1 + ρ2) . ≤

6 By Lemma 3.1 we have T T 2 e De e D( (D)) De ρ = + T † 2(nm)2 4(nm)2 eT D2e eT D1e = n n + m m 2n2 2m2 T T T T + (mem en D2 + emD1 nen )(Em (D2)+ (D1) En)†... { ⊗ ⊗ ⊗T 2 T ⊗ ...(me D2e + D1e ne ) /4(nm) m ⊗ n m ⊗ n } eT D2e eT D1e = n n + m m 2n2 2m2 T T T T Em En + (me e D2 + e D1 ne )( ( (D2))† + ( (D1))† )... { m ⊗ n m ⊗ n m2 ⊗ T T ⊗ n2 2 ...(mem D2en + D1em nen) /4(nm) T T ⊗ ⊗ } en D2en emD1em 2 T = + + m e D2( (D2))†D2e 2n2 2m2 { n T n 2 T 2 +n emD1( (D1))†D1em /(4(nm) 2 2 T } = ρ2 + ρ1

✷

Example 4.1 Let Gn = (gij) be the Manhattan distance matrix of a rectangular grid consisting of 1 row and n columns. Then g = i j . ij | − | 1 n n 1 i Let q ,...,q be the points in R − such that the ﬁrst i 1 coordinates of q are 1’s − 1 and the remaining n i coordinates are 0’s. Then it immediately follows that q ,...,qn 1 − 1 1 2 generate G and q ,...,qn lie on a hypersphere of radius ρ = (n 1) / and centered n 2 − at b = (1/2, 1/2,..., 1/2)T . Thus G is a spherical EDM of embedding dimension n 1. n − Note that in this example, we don’t assume that the origin coincides with the centroid of the points q1,...,qn in order to keep the expressions of the qi’s and b simple.

Now consider a rectangular grid of m rows and n columns. Let dˆij,kl be the Manhat- tan distance between the grid point of coordinates (i, j) and the grid point of coordinates (k, l). Then dˆ = i k + j l . ij,kl | − | | − | Let s = i + n(j 1) for j = 1,...,m and i = 1,...,n; and let t = k + n(l 1) for − − l = 1,...,m and k = 1,...,n. Then j = s/n , i = s n( s/n 1) and l = t/n , k = t n( t/n 1). ⌈ ⌉ − ⌈ ⌉− ⌈ ⌉ − ⌈ ⌉−

7 Deﬁne the nm nm matrix D = (d ) such that d = dˆ . Then it follows [8] × st st ij,kl that D = E G + G E , (8) m ⊗ n m ⊗ n where G is as deﬁned in Example 4.1. Equation (8) follows since (E G ) = i k n m ⊗ n st | − | where i = s n( s/n 1) and k = t n( t/n 1); and since (G E ) = j l − ⌈ ⌉− − ⌈ ⌉− m ⊗ n kl | − | where j = s/n and l = t/n . ⌈ ⌉ ⌈ ⌉ Thus we have the following two corollaries.

Corollary 4.1 Let D be the mn mn Manhattan distance matrix of a rectangular × grid of m rows and n columns. Then D is a spherical EDM of embedding dimension n + m 2. Furthermore, the points that generate D lie on a hypersphere of radius 1 − 1 2 ρ = (n + m 2) / . 2 − Proof. This follows from Theorems 4.1 and 4.2. ✷

Corollary 4.2 (Mittlemann and Peng [8, Theorem 2.6]) Let D be the mn mn × Manhattan distance matrix of a rectangular grid of m rows and n columns. Then (n + m 2)E /2 D 0. − mn − Example 4.2 Consider the Hamming distance matrix of the r-dimensional hypercube r r 1 2r Qr whose vertices are the 2 points in R with coordinates equal to 1 or 0. Let p ,...,p r r be the vertices of Qr and let D = (dij) be the 2 2 matrix where dij is the Ham- i j r × i j ming distance between p and p ; i.e., dij = k=1 pk pk . Then dij is also equal r i j 2 i j 2 | − | to k=1(pk pk) = p p . Therefore, DPis an EDM. Furthermore, the points 1 2r − || − || 1 1/2 p ,...,pP lie on a hypersphere, centered at their centoid, of radius ρ = 2 r . Thus D is a regular EDM of embedding dimension r.

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