2.3 Structures of Random Block Matrices
2.3.3 Block Tri-Diagonal (BTD) Matrices
We refer to an n×m matrix M as a BTD matrix if it can be written in the following form M= M1(n1,1 :e1) M2(n2, s2 :e2) .. . ML(nL, sL:m)
where s1 = 1, ei−2 < si ≤ si+1,ei ≤ei+1 and eL =m. Fig. 2.1c shows an example of
a BTD matrix for L= 3. In order to enumerate all full-rank BTD matrices for a given set of parameters, we will revisit and extend (2.2), so that the constraints of the BTD structure are incorporated. In an effort to facilitate the analysis, we will first discuss two relevant lemmata and introduce the notation (Φ1, . . . ,ΦL) to represent the horizontal
concatenation ofL matrices.
Lemma 2.4. Let M= (Φ1,Φ2)∈Fnq×(m1+m2) be a random matrix that has been con-
structed by horizontally concatenating Φ1 ∈ Fnq×m1 and Φ2 ∈ Fnq×m2. The number of
full-rank matrix realizations of Mcan be expressed as
γ(n, m1+m2) =γ(n, m1)γ(n−m1, m2)qm1m2 (2.12)
or
γ(n, m1+m2) =γ(n−m2, m1)γ(n, m2)qm1m2 (2.13)
Proof. Matrix M has full rank if its m1 +m2 columns are linearly independent or,
equivalently, m1 +m2 out of the n rows are linearly independent. This implies that
Φ1 and Φ2 should be not only full-rank matrices but their columns should span non-
overlapping vector subspaces. ForΦ1 to have full rank,m1of its rows should be linearly
independent. As we have already seen, there exist γ(n, m1) full-rank random matrices
of dimensions n×m1. For Φ2 to have full rank, m2 of its rows should also be linearly
independent. However, the columns ofΦ1 andΦ2 will span non-overlapping subspaces,
only if the m2 linearly independent rows of Φ2 are connected not to the m1 linearly
independent rows of Φ1 but to the remainingn−m1 rows. Therefore, the number of
full-rank realizations of Φ2 is equal to the number of full-rank (n−m1)×m2 random
matrices multiplied by the number of arbitrarily defined elements in the remainingm1
rows of Φ2. The former quantity is given by γ(n−m1, m2) and the latter quantity is
equal to qm1m2. This concludes the proof of (2.12). The same line of reasoning can be
followed to derive (2.13) if we first considerΦ2and then compute the number of possible
realizations ofΦ1.
Lemma 2.5. Let M = (Φ1,Φ2) ∈ Fqn×(m1+m2) be a BTD matrix, where Φ1 ∈ Fnq×m1
and Φ2 ∈ Fnq×m2. The structure of M and the dimensions of its sub-matrices are as
follows
and n=n1+n2+n3 ≥m1+m2. The number of full-rank matrices that have the same
structure as M is γ(M) =X r1 X r2 2 Y i=1 γri(ni+1, wi)γ(ni−ri−1, mi−ri)q ϕi (2.14)
where max(0, mi−ni +ri−1) ≤ ri ≤ min(ni+1, wi) and ϕi = (mi−ri)wi−1+niri for
i= 1,2 while w0 = 0.
Proof. As explained in Lemma 2.4, M will be a full-rank matrix if both Φ1 and Φ2
have full rank and their columns span non-overlapping subspaces. Observe that Φ1
can be transformed into a BLT matrix and Corollary 2.3 can be invoked to compute the number of full-rank matrices that have the structure of Φ1. If Φ(3)1 contains r1
linearly independent columns, the remainingm1−r1 columns of (Φ(1)1 ,Φ(2)1 ) should also
of full-rank realizations ofΦ1 can be obtained by
γΦ1 =γr1(n2, w1)γ(n1, m1−r1)q
n1r1
where max(0, m1−n1) ≤ r1 ≤ min(n2, w1). When the rank of Φ(3)1 is r1, the number
of linearly independent rows of Φ(3)
1 is also r1. Therefore, as per Lemma 2.4, n2−r1
rows of (Φ(1)
2 ,Φ
(2)
2 ) should only be considered in the enumeration of all valid full-rank
realizations ofΦ2 given by
γΦ2=γr2(n3, w2)γ(n2−r1, m2−r2)q
(m2−r2)w1qn2r2
where min(0, m2−n2+r1)≤r2≤max(n3, w2). If the productγΦ1γΦ2 is summed over
all values ofr1 and r2, expression (2.14) is obtained.
Proposition 2.6. A BTD matrix M= (M1;. . .;ML)∈Fnq×m has been built by verti-
cally concatenating Mi ∈Fnqi×m for i= 1, . . . , L, where n=n1+. . .+nL. For si ≤ei,
let all elements of Mi in columns si, ei and in-between take values from Fq while the
remaining columns of Mi consist of zeros. As per the BTD structure requirements,
we have s1 = 1, ei−2 < si ≤ si+1, ei ≤ ei+1 and eL = m. The number of full-rank
realizations of M is given by γBTD(M)= X r1 ···X rL−1 L Y i=1 γri(ni+1, wi)γ(ni−ri−1, mi−ri)q ϕi (2.15) where:
mi=ei−ei−1 for i= 2, . . . , L and m1=e1 for i= 1,
wi=ei−si+1+ 1 for i= 1, . . . , L−1 and wL= 0 for i=L, ϕi = (mi−ri)wi−1+niri,
n=n1+n2+. . .+nL≥m
and ri, for i= 1, . . . , L−1, takes values in the range
ri ≥max(0, mi−ni+ri−1) and
ri ≤min(ni+1, wi), while r0=rL= 0.
Proof. The BTD matrixMcan be rewritten as a horizontal concatenation ofLmatrices (Φ1, . . . ,ΦL), where Φi = 0i; Φ(1)i ,Φ (2) i ; 0i,Φ(3)i ;0i
for i= 1, . . . , L−1. Using the notation of Lemma 2.5, the dimensions of the random
matricesΦ(1)
i ,Φ
(2)
i andΦ
(3)
i areni×(mi−wi), ni×wi and ni+1×wi, respectively. On
the other hand, the dimensions of the zero matrices0i,0i and 0i are ( Pi−1
k=1nk)×mi,
structure
ΦL= 0L;Φ
(1)
L
where 0L is the (n−nL)×mL zero matrix and Φ
(1)
L is an nL×mL random matrix.
If we consider the first L−1 sub-matrices only, the number of full-rank matrices with structure (Φ1, . . . ,ΦL−1) is γ(Φ1,...,ΦL−1)=X r1 ···X rL−1 L−1 Y i=1 γri(ni+1, wi)γ(ni−ri−1, mi−ri)q ϕi (2.16)
as per Lemma2.5. The inclusion of the last sub-matrixΦLwill increase the number of
full-rank realisations of Mby a factor of γΦL, where
γΦL=γ(nL−rL−1, mL)q
mLwL−1. (2.17)
The product of (2.16) and (2.17) gives (2.15). Notice that (2.17) can be incorporated into (2.16) if we change the upper limit of the summation index ifrom L−1 to L and set rL= 0.
This section focused on random block matrices and demonstrated that Proposition 2.2
can be used to compute the number of full-rank BD matrices but, as Corollary 2.3
explained, it can also be extended to the case of BLT matrices. Proposition 2.6 was introduced for the enumeration of full-rank BTD matrices. The following section will discuss how the analysis of random block matrices can be used in the performance assessment of practical network coding techniques.