Determinants of block tridiagonal matrices
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).
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ABSTRACT: The formulas presented in [Molinari, L.G. Determinants of block tridiagonal matri-ces. Linear Algebra Appl., 2008; 429, 2221–2226] for evaluating the determinant of block tridiagonal matrices with (or without) corners are used to derive the determinant of any multidiagonal matri-ces with (or without) corners with some specified non-zero minors. Algorithms for calculation the determinant based on this method are given and properties of the determinants are studied. Some applications are presented.The electronic journal of linear algebra ELA 11/2012; 25:101-117. · 0.89 Impact Factor
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ABSTRACT: A Krawtchouk polynomial is introduced as the classical Mac-Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the -number and the -number, which are more generalized notions of the Krawtchouk polynomial in the P-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eberlein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.Journal of the Korean Mathematical Society 01/2013; 50(3). · 0.32 Impact Factor
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ABSTRACT: Recently, three computational algorithms for evaluating the determinant of quasi penta-diagonal matrices have been proposed by El-Mikkawy and Rahmo (Comput Math Appl 59:1386–1396, 2010), by Neossi Nguetchue and Abelman (Appl Math Comput 203:629–634, 2008), and by Jia et al. (Int J Comput Math 89:851–860, 2013), respectively. In the current paper, two novel algorithms with less computational costs are proposed for the determinant evaluation of general quasi penta-diagonal matrices and quasi penta-diagonal Toeplitz matrices. Furthermore, three numerical experiments are given to show the performance of our algorithms. All of the numerical computations were performed on a computer with aid of programs written in MATLAB.Journal of Mathematical Chemistry 01/2013; · 1.23 Impact Factor
arXiv:0712.0681v3 [math-ph] 16 Jun 2008
DETERMINANTS OF BLOCK
Luca Guido Molinari
Dipartimento di Fisica, Universit` a degli Studi di Milano,
and INFN, Sezione di Milano, Via Celoria 16, Milano, Italy
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
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, and decay rates of their matrix elements have been
investigated. 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 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 and Sogabe. These results encouraged me to
re-examine a nice identity that I derived in the context of transport, and
Email address: firstname.lastname@example.org (Luca Guido Molinari).
Preprint submitted to Elsevier16 June 2008
extend it as a mathematical result for general block-tridiagonal complex ma-
For ordinary tridiagonal matrices, determinants can be evaluated via multi-
plication of 2 × 2 matrices:
= (−1)n+1(bn···b1+ cn−1···c0)
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:
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:
where Imis the m× m unit matrix. The transfer matrix is nonsingular, since
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
(k = 2,...,n − 1)
The equations (7) are recursive and can be put in the form
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
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]
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).
is introduced for all factors in the transfer matrix T and produces intermediate
bkI2that commute, and allow us to simplify the determinant of the
− (−1)nz1+ z2
b1···bn[(z1+ z2) − (−1)n(b1···bn+ c0···cn−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
Multiplication of Lemma 1 by the previous equation, with parameter 1/z,
gives another variant:
detM(z)detM(1/z) = det
T + T−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. 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  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)
11is the upper left block of size m × m of the transfer matrix
Proof: The linear system M(0)Ψ = 0 can be translated into the following
equation, via the transfer matrix technique:
× ... (11)
Right multiplication by the nonsingular matrix
and rewriting the right-hand vector as the product
transform (11) into an equation for the transfer matrix T(0), that connects the
boundary components with ψn+1= 0 and ψ0= 0:
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
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 proposed a different procedure for the evaluation of the
Λk= Ak− Ck−1Λ−1
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 − 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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