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arXiv:0712.0681v3 [math-ph] 16 Jun 2008
DETERMINANTS OF BLOCK
TRIDIAGONAL MATRICES
Luca Guido Molinari
Dipartimento di Fisica, Universit` a degli Studi di Milano,
and INFN, Sezione di Milano, Via Celoria 16, Milano, Italy
Abstract
An identity is proven that evaluates the determinant of a block tridiagonal matrix
with (or without) corners as the determinant of the associated transfer matrix (or
a submatrix of it).
Key words: Block tridiagonal matrix, transfer matrix, determinant
1991 MSC: 15A15, 15A18, 15A90
1Introduction
A tridiagonal matrix with entries given by square matrices is a block tridi-
agonal matrix; the matrix is banded if off-diagonal blocks are upper or lower
triangular. Such matrices are of great importance in numerical analysis and
physics, and to obtain general properties is of great utility. The blocks of
the inverse matrix of a block tridiagonal matrix can be factored in terms of
two sets of matrices[10], and decay rates of their matrix elements have been
investigated[14]. While the spectral properties of tridiagonal matrices have
been under study for a long time, those of tridiagonal block matrices are at a
very initial stage[1,2].
What about determinants? A paper by El-Mikkawy[4] on determinants of
tridiagonal matrices triggered two interesting generalizations for the evalua-
tion of determinants of block-tridiagonal and general complex block matrices,
respectively by Salkuyeh[15] and Sogabe[17]. These results encouraged me to
re-examine a nice identity that I derived in the context of transport[11], and
Email address: luca.molinari@mi.infn.it (Luca Guido Molinari).
Preprint submitted to Elsevier16 June 2008
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extend it as a mathematical result for general block-tridiagonal complex ma-
trices.
For ordinary tridiagonal matrices, determinants can be evaluated via multi-
plication of 2 × 2 matrices:
det
a1 b1
c1...
c0
...
... bn−1
...
bn
cn−1 an
= (−1)n+1(bn···b1+ cn−1···c0)
+tr
an−bn−1cn−1
10
···
a2−b1c1
10
a1−bnc0
10
(1)
det
a1 b1
c1... ...
... bn−1
...
cn−1 an
=
an−bn−1cn−1
10
···
a2−b1c1
10
a10
1 0
11
(2)
Do these procedures generalize to block-tridiagonal matrices? The answer is
affirmative. If the matrix has corner blocks, the determinant is proportional
to that of an associated transfer matrix, in general of much smaller size. The
proof is simple and is given in section 2. A simple modification yields a formula
for the determinant when corner blocks are absent, and is given in section 3.
The relation with Salkuyeh’s recursion formula is then shown.
2 The Duality Relation
Consider the following block-tridiagonal matrix M(z) with blocks Ai, Biand
Ci−1 (i = 1,...,n) that are complex m × m matrices. It is very useful to
introduce also a complex parameter z in the corner blocks:
M(z) =
A1 B1
1
zC0
C1
...
...
...
... Bn−1
zBn
Cn−1 An
(3)
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It is required that off-diagonal blocks are nonsingular: detBi?= 0 and detCi−1?=
0 for all i. As it will be explained, the matrix is naturally associated with a
transfer matrix, built as the product of n matrices of size 2m × 2m:
T =
−B−1
nAn−B−1
nCn−1
Im
0
...
−B−1
1A1−B−1
1 C0
Im
0
(4)
where Imis the m× m unit matrix. The transfer matrix is nonsingular, since
detT =
n
?
i=1
det[B−1
iCi−1] (5)
The main result, the duality relation, relies on the following lemma:
Lemma 1detM(z) =(−1)nm
(−z)mdet[T − z I2m] det[B1...Bn]
Proof: The equation M(z)Ψ = 0 has a nontrivial solution provided that
detM(z) = 0, and corresponds to the following linear system in terms of
the blocks of the matrix and the components ψk∈ Cmof the null vector Ψ:
A1ψ1+ B1ψ2+ z−1C0ψn = 0
Bkψk+1+ Akψk+ Ck−1ψk−1= 0
z Bnψ1+ Anψn+ Cn−1ψn−1 = 0
(6)
(7)
(8)
(k = 2,...,n − 1)
The equations (7) are recursive and can be put in the form
ψk+1
ψk
=
−B−1
kAk−B−1
kCk−1
Im
0
ψk
ψk−1
and iterated. Inclusion of the boundary equations (6) and (8) produces an
eigenvalue equation for the full transfer matrix (4) that involves only the end
vector-components:
T
ψ1
1
zψn
= z
ψ1
1
zψn
(9)
Equation (9) has a nontrivial solution if and only if det[T − zI2m] = 0, which
is dual to the condition detM(z) = 0. Both zmdetM(z) and det[T − zI2m]
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are polynomials in z of degree 2m and share the same roots, which can-
not be zero by (5). Therefore, the polynomials coincide up to a constant
of proportionality, which is found by considering the limit case of large z:
detM(z) ≈ (−1)nm(−z)mdet[B1···Bn]. ?
Before proceeding, let us show that in the special case of tridiagonal matrices
with corners (m = 1), Lemma 1 with z = 1 yields (1).
The factorization
−ak−1
bk−1−ck−2
bk−1
10
=
−
1
bk−10
01
ak−1−ck−2bk−2
10
10
0 −
1
bk−2
(10)
is introduced for all factors in the transfer matrix T and produces intermediate
factors
lemma:
1
bkI2that commute, and allow us to simplify the determinant of the
det
−an
bn−cn−1
bn
10
···
−a1
b1−c0
b1
10
− I2
= det
(−1)n−1
b1···bn−1
−1
an−bn−1cn−1
− (−1)nz1+ z2
b1···bn
bn0
01
an−bn−1cn−1
10
···
a1−bnc0
10
10
0 −1
bn
− I2
=
1
b2
1···b2
n
det
10
···
a1−bnc0
10
− (−1)nb1···bnI2
=
z1z2
b2
= −(−1)n
b1···bn[(z1+ z2) − (−1)n(b1···bn+ c0···cn−1)]
1···b2
n
+ 1
z1and z2are the eigenvalues of the transfer matrix in (1), whose trace is z1+z2
and whose determinant is z1z2= (b1···bn)(c0···cn−1). ?
Multiplication of Lemma 1 by detT−1gives a variant of it:
detM(z) = (−1)nm(−z)mdet(T−1−1
z) det[C0...Cn−1]
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Multiplication of Lemma 1 by the previous equation, with parameter 1/z,
gives another variant:
detM(z)detM(1/z) = det
?
T + T−1−
?
z +1
z
??
det[B1C0...BnCn−1]
Instead of M(z), consider the matrix M(z) − λInm and the corresponding
transfer matrix T(λ) obtained by replacing the entries Aiwith Ai−λIm. Then
Lemma 1 has a symmetric form, where the roles of eigenvalue and parameter
exchange between the matrices. For this reason it is called a duality relation.
Theorem 1 (The Duality Relation)
det[λInm− M(z)] = (−z)−mdet[T(λ) − zI2m] det[B1···Bn]
It shows that the parameter z, which enters in M(z) as a boundary term,
is related to eigenvalues of the matrix T(λ) that connects the eigenvector of
M(z) at the boundaries.
The duality relation was initially obtained and discussed for Hermitian block
matrices[11,12,13]. For n = 2 it is due to Lee and Ioannopoulos[9]. Here I have
shown that it holds for generic block-tridiagonal matrices, and the proof given
is even simpler. The introduction of corner values z and 1/z in Hermitian
tridiagonal matrices (ck = b∗
model for vortex depinning in superconductors, as a tool to link the decay of
eigenvectors to the permanence of corresponding eigenvalues on the real axis.
It has been a subject of intensive research[16,5,6,18]. The generalization to
block matrices is interesting for the study of transport in discrete structures
such as nanotubes or molecules[8,3,19].
k) was proposed by Hatano and Nelson [7] in a
3 Block tridiagonal matrix with no corners
By a modification of the proof of the lemma, one obtains an identity for the
determinant of block-tridiagonal matrices M(0)with no corners (Bn= C0= 0
in the matrix (3)):
Theorem 2detM(0)= (−1)nmdet[T(0)
11] det[B1···Bn−1]
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where T(0)
11is the upper left block of size m × m of the transfer matrix
T(0)=
−An−Cn−1
Im
0
−B−1
n−1An−1−B−1
n−1Cn−2
Im
0
...
−B−1
1A1−B−1
1
Im
0
Proof: The linear system M(0)Ψ = 0 can be translated into the following
equation, via the transfer matrix technique:
ψn
−C−1
n−1Anψn
=
−B−1
n−1An−1−B−1
n−1Cn−2
Im
0
× ...(11)
×
−B−1
2 A2−B−1
2C1
Im
0
−B−1
1 A1ψ1
ψ1
Right multiplication by the nonsingular matrix
−An−Cn−1
Im
0
and rewriting the right-hand vector as the product
−B−1
1A1−B−1
1
Im
0
ψ1
0
transform (11) into an equation for the transfer matrix T(0), that connects the
boundary components with ψn+1= 0 and ψ0= 0:
0
ψn
= T(0)
ψ1
0
(12)
Equation (12) implies that detT(0)
implication translates into an identity by introducing the parameter λ and
comparing the polynomials det[λInm− M(0)] and detT(0)(λ) (obtained by re-
placing blocks Aiwith Ai− λIm). Since both are polynomials in λ of degree
nm and with the same roots, they must be proportional. Their behaviour for
large λ fixes the constant. ?
11= 0, which is dual to detM(0)= 0. The
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For tridiagonal matrices (m = 1) blocks are just scalars and, by means of (10),
one shows Theorem 2 simplifies to (2).
The formula for the evaluation of detM(0)requires n−1 inversions B−1
tiplication of n matrices of size 2m × 2m, and the final evaluation of a deter-
minant. Salkuyeh[15] proposed a different procedure for the evaluation of the
same determinant:
k, mul-
detM(0)=
n
?
k=1
detΛk
Λk= Ak− Ck−1Λ−1
k−1Bk−1,Λ1= A1
It requires n − 1 inversions of matrices of size m × m, and the evaluation of
their determinants. I show that the two procedures are related.
The transfer matrix T(0)= T(n) is the product of n matrices. Let T(k) be the
partial product of k matrices. Then:
T(k) =
−B−1
kAk−B−1
kCk−1
Im
0
T(k − 1)
This produces a two-term recurrence relation for blocks
T(k)11 = −B−1
kAkT(k − 1)11− B−1
kCk−1T(k − 2)11
with T(1)11= −B−1
for Λk= −BkT(k)11[T(k − 1)11]−1.
1A1and T(0)11= Im. The equations by Salkuyeh result
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