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Some comments on tridiagonal (p,r)-Toeplitz matrices
Linear Algebra and its Applications 572 (2019) 46–50
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Some comments on tridiagonal (p, r)-Toeplitz matrices Milica Anđelić a,∗ , Carlos M. da Fonseca b,c a
Department of Mathematics, Kuwait University, Safat 13060, Kuwait Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait c University of Primorska, FAMNIT, Glagoljsaška 8, 6000 Koper, Slovenia b
a r t i c l e
i n f o
Article history: Received 29 January 2019 Accepted 1 March 2019 Available online 7 March 2019 Submitted by R. Brualdi
a b s t r a c t The aim of this note is to extend the notion of a tridiagonal (p, r)-Toeplitz matrix recently introduced and to provide procedures for short proofs to results on the invertibility of such matrices. © 2019 Elsevier Inc. All rights reserved.
MSC: 15A18 65F15 15B05 42C05 Keywords: Eigenvalues Tridiagonal k-Toeplitz matrices Tridiagonal (p, r)-Toeplitz matrices Chebyshev polynomials of second kind
* Corresponding author. E-mail addresses: [email protected] (M. Anđelić), [email protected], [email protected] (C.M. da Fonseca). https://doi.org/10.1016/j.laa.2019.03.001 0024-3795/© 2019 Elsevier Inc. All rights reserved.
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
47
1. Introduction Recently, A.M. Encinas and M.J. Jiménez in [2, Definition 4.1] introduced the notion of an n × n tridiagonal (p, r)-Toeplitz matrix, for a positive integer p and a nonzero real number r, such that n ≡ 0 (mod p). A tridiagonal (p, r)-Toeplitz matrix is seen as a tridiagonal matrix A = (aij ) such that ai+p,j+p = raij ,
for i, j = 1, 2, . . . , n − p ,
(1.1)
that is ⎛
A(p,r) n
a1
⎜ ⎜ c1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b1 .. . .. .
⎞ ..
.
ap cp
bp ra1 rc1
rb1 .. . .. .
..
.
rap rcp
rbp r 2 a1 r2 c1
r2 b1 .. . .. .
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟ .⎟ ⎠
.
(1.2)
n×n
We ignore the trivial cases and we read all the non-mentioned entries as zero. For r = 1 we get what is known as a tridiagonal p-Toeplitz (or k-Toeplitz) matrix, with p n, while for p = r = 1, we get a tridiagonal Toeplitz matrix. In [2], the authors provide necessary and sufficient conditions for the invertibility based on an elaborated technique of boundary value problems associated to second order linear difference equations which can be expressed in terms of the discrete Schrödinger operator. In this short note, we recall known results in the literature about the invertibility of a tridiagonal k-Toeplitz matrix, extend the notion of a tridiagonal (p, r)-Toeplitz matrix, and provide short proofs for the main results in [2]. 2. Invertible tridiagonal k-Toeplitz matrices The concept of tridiagonal k-Toeplitz matrix goes back to 1985, with the paper [6] by M.J.C. Gover and S. Barnett. The particular case when k = 2 and the entries of the two subdiagonals are all 1’s, was considered in 1953 by J.F. Elliott [1, Section IV.4]. The spectrum of a general tridiagonal 2-Toeplitz matrix was first studied by M.J.C. Gover in 1994 [5]. Further results followed soon after by F. Marcellán and J. Petronilho [7]. In 1999, C.M. da Fonseca and J. Petronilho [4] also considered the case when k = 3 and,
48
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
later on, the characteristic polynomial of a tridiagonal k-Toeplitz matrix was found, for any k, when analysing the invertibility conditions for such matrix [3]. Namely, for p = 2 in (1.2), we have (cf. [4, Theorem 4.1]) (2,1)
det A2n
a1 a2 − b1 c1 − b2 c2 √ b1 b2 c1 c2 )n Un 2 b1 b2 c1 c2 √
a1 a2 − b1 c1 − b2 c2 b2 c2 √ +√ Un−1 b1 c1 2 b1 b2 c1 c2
=(
(2.1)
and, otherwise, (2,1) det A2n+1
= a1 (
n
b1 b2 c1 c2 ) Un
a1 a2 − b1 c1 − b2 c2 √ 2 b1 b2 c1 c2
,
(2.2)
where Un (x) =
sin(n + 1)θ sin θ
with x = cos θ
(0 θ < π)
is the Chebyshev polynomial of second kind of degree n. For p > 2, we need a deeper approach which can be found in [3, Section 5]. Namely, for j i, we define the polynomial Δi,j of degree j − i + 1 in x by ⎛ ⎜ ⎜ Δi,j (x) = det ⎜ ⎜ ⎝
x − ai ci
bi .. . .. .
⎞ ..
.
..
.
bj−1 x − aj
cj−1
⎟ ⎟ ⎟, ⎟ ⎠
and, for j < i, set Δi,j (x) =
0 if j < i − 1 . 1 if j = i − 1
Moreover, we set ϕp (x) =
1 (Dp (x) + (−1)p (b1 c1 · · · bp−1 cp−1 + bp cp )) , 2μ
a polynomial of degree p, where Dp is a monic polynomial also of degree p, ⎛
x − a1 ⎜ b c ⎜ 1 1 ⎜ ⎜ Dp (x) = ⎜ ⎜ ⎜ ⎝ bp cp
1 x − a2 b2 c2
1 1 ..
.
..
..
.
..
.
. bp−1 cp−1
1 x − ap
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
49
and
μ=
b1 · · · bp c1 · · · cp .
Thus, if n ≡ s (mod p), with 0 s p − 1, then
det An(p,1)
n
n/p
= (−1) μ
Δ1,s (0) Un/p (ϕp (0))
s bp cp + bi ci Δs+2,p−1 (0) U(n−p)/p (ϕp (0)) . μ i=1
(2.3)
(p,1)
Since all quantities in (2.3) can be easily computed, checking the singularity of An immediate.
is
3. The invertibility of tridiagonal (p, r)-Toeplitz matrices (p,r)
Our first observation regarding the tridiagonal (p, r)-Toeplitz matrices An as defined in [2], is that we can extend the concept to any order n. We recall that in [2] all results are confined to n ≡ 0 (mod p). Actually, we can go far beyond that and consider (1.1) for any matrix, not necessarily tridiagonal. However, we believe that in this general case the invertibility analysis is much harder to tackle. Of course, all matrices can be complex as well. Perhaps more relevant is the fact that the main results in [2] can be immediately derived from [3,4] and, indeed, extended. For that purpose we should consider the invertible diagonal matrix of order n given by
Mn = diag (1, . . . , 1, r, . . . , r, r2 , . . .) .
p×
p×
It is straightforward to check that
Mn A˜n(p,1) = A(p,r) , n where
50
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
⎛
A˜n(p,1)
a1
⎜ ⎜c ⎜ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b1 .. . .. .
⎞ ..
.
..
.
cp−1
bp−1 ap cp r
bp a1 c1
b1 .. .
..
..
..
.
.
. cp−1
bp−1 ap cp r
bp a1 c1
b1 .. . ..
.
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟ .⎟ ⎠
.
n×n
(p,1) Now, we only need to consider the matrix A˜n and apply the results described in Section 2, with n ≡ 0 (mod p), and we obtain the main results one can find in [2]. As we mentioned above, our approach is absolutely general, since it can be done for any order n. The determinant of Mn can be easily determined. (p,r) As for the inverse of An , we just need to take into account the straightforward fact
Mn−1 = diag (1, . . . , 1, 1/r, . . . , 1/r, 1/r2 , . . .) ,
p×
p×
(p,1) and find the inverse of A˜n applying directly [4, Theorem 4.1], for p = 2, or [3, Theorem 4.1], for any p 3. An interesting problem to investigate is, for any n, to find the characteristic polynomial (p,r) of An . We leave it as an open problem for further discussion.
References [1] J.F. Elliott, The Characteristic Roots of Certain Real Symmetric Matrices, Master’s thesis, University of Tennessee, 1953. [2] A.M. Encinas, M.J. Jiménez, Explicit inverse of a tridiagonal (p, r)-Toeplitz matrix, Linear Algebra Appl. 542 (2018) 402–421. [3] C.M. da Fonseca, J. Petronilho, Explicit inverse of a tridiagonal k-Toeplitz matrix, Numer. Math. 100 (3) (2005) 457–482. [4] C.M. da Fonseca, J. Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra Appl. 325 (2001) 7–21. [5] M.J.C. Gover, The eigenproblem of a tridiagonal 2-Toeplitz matrix, Linear Algebra Appl. 197/198 (1994) 63–78. [6] M.J.C. Gover, S. Barnett, Inversion of Toeplitz matrices which are not strongly singular, IMA J. Numer. Anal. 5 (1985) 101–110. [7] F. Marcellán, J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997) 169–208.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Some comments on tridiagonal (p, r)-Toeplitz matrices Milica Anđelić a,∗ , Carlos M. da Fonseca b,c a
Department of Mathematics, Kuwait University, Safat 13060, Kuwait Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait c University of Primorska, FAMNIT, Glagoljsaška 8, 6000 Koper, Slovenia b
a r t i c l e
i n f o
Article history: Received 29 January 2019 Accepted 1 March 2019 Available online 7 March 2019 Submitted by R. Brualdi
a b s t r a c t The aim of this note is to extend the notion of a tridiagonal (p, r)-Toeplitz matrix recently introduced and to provide procedures for short proofs to results on the invertibility of such matrices. © 2019 Elsevier Inc. All rights reserved.
MSC: 15A18 65F15 15B05 42C05 Keywords: Eigenvalues Tridiagonal k-Toeplitz matrices Tridiagonal (p, r)-Toeplitz matrices Chebyshev polynomials of second kind
* Corresponding author. E-mail addresses: [email protected] (M. Anđelić), [email protected], [email protected] (C.M. da Fonseca). https://doi.org/10.1016/j.laa.2019.03.001 0024-3795/© 2019 Elsevier Inc. All rights reserved.
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
47
1. Introduction Recently, A.M. Encinas and M.J. Jiménez in [2, Definition 4.1] introduced the notion of an n × n tridiagonal (p, r)-Toeplitz matrix, for a positive integer p and a nonzero real number r, such that n ≡ 0 (mod p). A tridiagonal (p, r)-Toeplitz matrix is seen as a tridiagonal matrix A = (aij ) such that ai+p,j+p = raij ,
for i, j = 1, 2, . . . , n − p ,
(1.1)
that is ⎛
A(p,r) n
a1
⎜ ⎜ c1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b1 .. . .. .
⎞ ..
.
ap cp
bp ra1 rc1
rb1 .. . .. .
..
.
rap rcp
rbp r 2 a1 r2 c1
r2 b1 .. . .. .
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟ .⎟ ⎠
.
(1.2)
n×n
We ignore the trivial cases and we read all the non-mentioned entries as zero. For r = 1 we get what is known as a tridiagonal p-Toeplitz (or k-Toeplitz) matrix, with p n, while for p = r = 1, we get a tridiagonal Toeplitz matrix. In [2], the authors provide necessary and sufficient conditions for the invertibility based on an elaborated technique of boundary value problems associated to second order linear difference equations which can be expressed in terms of the discrete Schrödinger operator. In this short note, we recall known results in the literature about the invertibility of a tridiagonal k-Toeplitz matrix, extend the notion of a tridiagonal (p, r)-Toeplitz matrix, and provide short proofs for the main results in [2]. 2. Invertible tridiagonal k-Toeplitz matrices The concept of tridiagonal k-Toeplitz matrix goes back to 1985, with the paper [6] by M.J.C. Gover and S. Barnett. The particular case when k = 2 and the entries of the two subdiagonals are all 1’s, was considered in 1953 by J.F. Elliott [1, Section IV.4]. The spectrum of a general tridiagonal 2-Toeplitz matrix was first studied by M.J.C. Gover in 1994 [5]. Further results followed soon after by F. Marcellán and J. Petronilho [7]. In 1999, C.M. da Fonseca and J. Petronilho [4] also considered the case when k = 3 and,
48
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
later on, the characteristic polynomial of a tridiagonal k-Toeplitz matrix was found, for any k, when analysing the invertibility conditions for such matrix [3]. Namely, for p = 2 in (1.2), we have (cf. [4, Theorem 4.1]) (2,1)
det A2n
a1 a2 − b1 c1 − b2 c2 √ b1 b2 c1 c2 )n Un 2 b1 b2 c1 c2 √
a1 a2 − b1 c1 − b2 c2 b2 c2 √ +√ Un−1 b1 c1 2 b1 b2 c1 c2
=(
(2.1)
and, otherwise, (2,1) det A2n+1
= a1 (
n
b1 b2 c1 c2 ) Un
a1 a2 − b1 c1 − b2 c2 √ 2 b1 b2 c1 c2
,
(2.2)
where Un (x) =
sin(n + 1)θ sin θ
with x = cos θ
(0 θ < π)
is the Chebyshev polynomial of second kind of degree n. For p > 2, we need a deeper approach which can be found in [3, Section 5]. Namely, for j i, we define the polynomial Δi,j of degree j − i + 1 in x by ⎛ ⎜ ⎜ Δi,j (x) = det ⎜ ⎜ ⎝
x − ai ci
bi .. . .. .
⎞ ..
.
..
.
bj−1 x − aj
cj−1
⎟ ⎟ ⎟, ⎟ ⎠
and, for j < i, set Δi,j (x) =
0 if j < i − 1 . 1 if j = i − 1
Moreover, we set ϕp (x) =
1 (Dp (x) + (−1)p (b1 c1 · · · bp−1 cp−1 + bp cp )) , 2μ
a polynomial of degree p, where Dp is a monic polynomial also of degree p, ⎛
x − a1 ⎜ b c ⎜ 1 1 ⎜ ⎜ Dp (x) = ⎜ ⎜ ⎜ ⎝ bp cp
1 x − a2 b2 c2
1 1 ..
.
..
..
.
..
.
. bp−1 cp−1
1 x − ap
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
49
and
μ=
b1 · · · bp c1 · · · cp .
Thus, if n ≡ s (mod p), with 0 s p − 1, then
det An(p,1)
n
n/p
= (−1) μ
Δ1,s (0) Un/p (ϕp (0))
s bp cp + bi ci Δs+2,p−1 (0) U(n−p)/p (ϕp (0)) . μ i=1
(2.3)
(p,1)
Since all quantities in (2.3) can be easily computed, checking the singularity of An immediate.
is
3. The invertibility of tridiagonal (p, r)-Toeplitz matrices (p,r)
Our first observation regarding the tridiagonal (p, r)-Toeplitz matrices An as defined in [2], is that we can extend the concept to any order n. We recall that in [2] all results are confined to n ≡ 0 (mod p). Actually, we can go far beyond that and consider (1.1) for any matrix, not necessarily tridiagonal. However, we believe that in this general case the invertibility analysis is much harder to tackle. Of course, all matrices can be complex as well. Perhaps more relevant is the fact that the main results in [2] can be immediately derived from [3,4] and, indeed, extended. For that purpose we should consider the invertible diagonal matrix of order n given by
Mn = diag (1, . . . , 1, r, . . . , r, r2 , . . .) .
p×
p×
It is straightforward to check that
Mn A˜n(p,1) = A(p,r) , n where
50
M. Anđelić, C.M. da Fonseca / Linear Algebra and its Applications 572 (2019) 46–50
⎛
A˜n(p,1)
a1
⎜ ⎜c ⎜ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b1 .. . .. .
⎞ ..
.
..
.
cp−1
bp−1 ap cp r
bp a1 c1
b1 .. .
..
..
..
.
.
. cp−1
bp−1 ap cp r
bp a1 c1
b1 .. . ..
.
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟ .⎟ ⎠
.
n×n
(p,1) Now, we only need to consider the matrix A˜n and apply the results described in Section 2, with n ≡ 0 (mod p), and we obtain the main results one can find in [2]. As we mentioned above, our approach is absolutely general, since it can be done for any order n. The determinant of Mn can be easily determined. (p,r) As for the inverse of An , we just need to take into account the straightforward fact
Mn−1 = diag (1, . . . , 1, 1/r, . . . , 1/r, 1/r2 , . . .) ,
p×
p×
(p,1) and find the inverse of A˜n applying directly [4, Theorem 4.1], for p = 2, or [3, Theorem 4.1], for any p 3. An interesting problem to investigate is, for any n, to find the characteristic polynomial (p,r) of An . We leave it as an open problem for further discussion.
References [1] J.F. Elliott, The Characteristic Roots of Certain Real Symmetric Matrices, Master’s thesis, University of Tennessee, 1953. [2] A.M. Encinas, M.J. Jiménez, Explicit inverse of a tridiagonal (p, r)-Toeplitz matrix, Linear Algebra Appl. 542 (2018) 402–421. [3] C.M. da Fonseca, J. Petronilho, Explicit inverse of a tridiagonal k-Toeplitz matrix, Numer. Math. 100 (3) (2005) 457–482. [4] C.M. da Fonseca, J. Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra Appl. 325 (2001) 7–21. [5] M.J.C. Gover, The eigenproblem of a tridiagonal 2-Toeplitz matrix, Linear Algebra Appl. 197/198 (1994) 63–78. [6] M.J.C. Gover, S. Barnett, Inversion of Toeplitz matrices which are not strongly singular, IMA J. Numer. Anal. 5 (1985) 101–110. [7] F. Marcellán, J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997) 169–208.