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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

References

[1] A. I. Aptekarev and E. M. Nikishin, The scattering problem for a discrete

Sturm-Liouville operator, Math. USSR Sbornik 49 (1984) 325–355.

[2] J. Br¨ uning, D. Chelkak and E. Korotyaev, Inverse spectral analysis for finite

matrix-valued Jacobi operators. Available from: ¡arXiv:math.SP/0607809¿.

[3] S. Compernolle, L. Chibotaru and A. Coulemans, Eigenstates and transmission

coefficients of finite-sized nanotubes, J. Chem. Phys. 119 (2003) 2854–2873.

[4] M. El-Mikkawy, A note on a three-term recurrence for a tridiagonal matrix,

Appl. Math. Comp. 139 (2003) 503–511.

7

Page 8

[5] J. Feinberg and A. Zee, Spectral curves of non-Hermitean Hamiltonians, Nucl.

Phys. B 552 [FS] (1999) 599-623.

[6] I. Ya. Goldsheid and B. Khoruzhenko, Distribution of eigenvalues in non-

Hermitian Anderson models, Phys. Rev. Lett. 80 (1998) 2897–2900.

[7] N. Hatano and D. R. Nelson, Localization transition in quantum mechanics,

Phys. Rev. Lett. 77 (1996) 570–573.

[8] T. Kostyrko, M. Bartkowiak and G. D. Mahan, Reflection by defects in a tight-

binding model of nanotubes, Phys. Rev. B 59 (1999) 3241–3249.

[9] D. H. Lee and J. D. Ioannopoulos, Simple scheme for surface-band calculations.

II. The Green’s function, Phys. Rev. B 23 (1981) 4997–5004.

[10] G. Meurant, A review on the inverse of symmetric tridiagonal and block

tridiagonal matrices, SIAM J. Matrix Anal. Appl. 13 (1992) 707–728.

[11] L. Molinari, Transfer matrices and tridiagonal-block Hamiltonians with periodic

and scattering boundary conditions, J. Phys. A: Math Gen. 30 (1997) 983–997.

[12] L. Molinari, Transfer matrices, non-hermitian Hamiltonians and resolvents:

some spectral identities, J. Phys. A: Math. Gen. 31 (1998) 8553–8562.

[13] L. G. Molinari, Spectral duality and distribution of exponents, J. Phys. A:

Math. Gen. 36 (2002) 4081–4090.

[14] E. D. Nabben, Decay rates of the inverse of nonsymmetric tridiagonal and band

matrices, SIAM J. Matrix Anal. Appl. 20 (1999) 820–837.

[15] D. K. Salkuyeh, Comments on “A note on a three-term recurrence for a

tridiagonal matrix”, Appl. Math. Comp. 176 (2006) 442–444.

[16] N. M. Schnerb and D. R. Nelson, Winding numbers, complex currents and

non-Hermitian localization, Phys. Rev. Lett. 80 (1998) 5172–5175.

[17] T. Sogabe, On a two-term recurrence for the determinant of a general matrix,

Appl. Math. Comp. 187 (2007) 785–788.

[18] L. N. Trefethen and M. Embree, Spectra and Pseudospectra. The Behaviour

of Nonnormal Matrices and Operators, Princeton University Press, Princeton,

2005.

[19] H. Yamada, Electronic localization properties of a double strand of DNA: a

simple model with long range correlated hopping disorder, Int. J. Mod. Phys.

B 18 (2004) 1697–1716.

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