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Characteristic polynomials and eigenvaluesfor a family of tridiagonal real symmetric
matrices
Emily [email protected]
Rita [email protected]
February 23, 2020
Left to right are authors Post, Gullerud’s head, and Mbirika.1
Abstract
We explore a certain family {An}∞n=1 of n × n tridiagonal real symmetric matri-ces. After deriving a three-term recurrence relation for the characteristic polynomialsof this family, we find a closed form solution. The coefficients of these characteristicpolynomials turn out to involve the diagonal entries of Pascal’s triangle in an attrac-tively inviting manner. Lastly, we explore a relation between the eigenvalues of variousmembers of the family. More specifically, we give a sufficient condition on the valuesm,n ∈ N for when spec(Am) is contained in spec(An).
1The authors thank the hospitality of author Post’s uncles Rick and Mike whose porch provided a stim-ulating environment on Memorial Day in 2017 for us to prove that Equations (2) and (3), shown on thewhiteboard in the picture above, satisfy the three-term recurrence relation in Theorem 3.2.
1
Contents
1 Introduction 2
2 Preliminaries and definitions 5
3 The characteristic polynomials of An 63.1 A recurrence relation for fn(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 The fn(λ) and their connection to Pascal’s triangle . . . . . . . . . . . . . . 73.3 A closed form fn(λ) for the recurrence relation . . . . . . . . . . . . . . . . . 93.4 Parity properties of the fn(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 The spectrum of An 134.1 A trigonometric closed form for the roots of fn(λ) . . . . . . . . . . . . . . . 154.2 Spectral radius of An and some bounds questions . . . . . . . . . . . . . . . 174.3 Sufficient condition for spec(Am) ⊂ spec(An) . . . . . . . . . . . . . . . . . 17
5 Open questions 195.1 Conjectures on fn(λ) = Fn+1 where Fn+1 is the (n+ 1)th Fibonacci number . 20
1 Introduction
Tridiagonal matrices arise in many science and engineering areas, for example in parallelcomputing, telecommunication system analysis, and in solving differential equations usingfinite differences [2, 7, 4]. In particular, the eigenvalues of symmetric tridiagonal matriceshave been studied extensively starting with Golub in 1962, and moreover, a search on Math-SciNet reveals that over 40 papers have been published on this topic alone since then [5].
Tridiagonal real symmetric matrices are a subclass of the class of real symmetric matrices.A real symmetric matrix is a square matrix A with real-valued entries that are symmetricabout the diagonal; that is, A equals its transpose AT. Symmetric matrices arise naturallyin a variety of applications. Real symmetric matrices in particular enjoy the following twoproperties: (1) all of their eigenvalues are real, and (2) the eigenvectors corresponding todistinct eigenvalues are orthogonal. In this paper, we investigate the family of real symmetricn× n matrices of the form
An =
0 1
1. . . . . .. . . . . . 1
1 0
with ones on the superdiagonal and subdiagonal and zeroes in every other entry. Thesematrices, in particular, arise as the adjacency matrices of the path graphs and hence arefundamental objects in the field of spectral graph theory.aBa: [CITE SOURCES!!!!] Thefollowing are the three main results of this paper:
2
• MAIN RESULT 1: We give a closed form expression for the characteristic polyno-mials fn(λ) of our An matrices.
• MAIN RESULT 2: We show that the coefficients of these fn(λ) polynomials havean intimate connection to the diagonals of Pascal’s triangle.
• MAIN RESULT 3: Utilizing a closed form expression for the set of eigenvalues ofour matrices, we give an upper bound on the spectral radius ρ(An) of An and providea sufficiency condition on the values m,n ∈ N that guarantees spec(Am) ⊂ spec(An).
We were initially drawn to this work when noticing the first few characteristic polynomialsof our An matrices. For example, we give the first seven polynomials below.
n-value fn(λ)
1 −1λ2 1λ2 −13 −1λ3 +2λ4 1λ4 −3λ2 +15 −1λ5 +4λ3 −3λ6 1λ6 −5λ4 +6λ2 −17 −1λ7 +6λ5 −10λ3 +4λ
Color-coding the coefficients to make the connection more transparent, we see that the fourcolumns of coefficients correspond directly to the the first four diagonals in Pascal’s triangle.We used this observation to conjecture and eventually prove the following formula for themth characteristic polynomial
fm(λ) =
bm2 c∑i=0
(−1)m+i
(m− ii
)λm−2i.
We prove that this formula satisfies the three-term recurrence formula
fn(λ) = −λfn−1(λ)− fn−2(λ)
with initial conditions f1(λ) = −λ and f2(λ) = λ2 − 1, thereby establishing our first mainresult. Our second main result uses a closed form for the eigenvalues of the matrix An toexplore some properties of the spectrum of An. Our third main result was motivated byan observation that the four distinct eigenvalues of the A4 matrix are the golden ratio, itsreciprocal, and their negatives. Furthermore via Mathematica, we observed that these foureigenvalues appear in the spectrum of the An matrices for the n-values
4, 9, 14, 19, 24, 29, 34, 39, and 44.
3
All of these numbers being congruent to 4 modulo 5 seemed to be no accident. Motivatedby this tantalizing observation, we use a closed form expression
λs = 2 cos
(sπ
m+ 1
)for s = 1, . . . , n
for the n distinct eigenvalues of each An matrix to give a sufficiency criterion for when theeigenvalues of the matrix Am are also eigenvalues of the matrix An, thereby establishing ourthird main result.
The breakdown of the paper is as follows. In Section 2, we give some preliminaries anddefinitions. In Section 3, we focus on the characteristic polynomials of the An matrices; inparticular,
(1) we derive a three-term recurrence relation for the characteristic polynomials of thefamily of matrices {An}∞n=1 in Subsection 3.1,
(2) we unveil a tantalizing connection between the coefficients of the family of charac-teristic polynomials {fn(λ)}∞n=1 of the matrices {An}∞n=1 and the diagonal columns ofPascal’s triangle in Subsection 3.2,
(3) we prove that our closed form satisfies the recurrence relation in Subsection 3.3, and
(4) we give a parity property of each fn(λ) dependent on the parity of n in Subsection 3.4.
In Section 4, we focus on the spectrum of the An matrices; in particular,
(5) we prove that a given trigonometric closed form yields the eigenvalues of each Anmatrix in Subsection 4.1,
(6) we investigate the spectral radius of An and in particular give lower and upper boundsof spec(An) in Subsection 4.2, and
(7) we prove a sufficiency condition on the values m,n ∈ N that guarantees the contain-ment spec(Am) ⊂ spec(An) in Subsection 4.3.
Finally in Section 5, we provide some open questions.
Acknowledgments
aBa: [I feel that these words are fine for the arXiv, as it is truthful and also in the spiritof this ’unfunded’ project. But we can change it to something boring/serious for submittedversion.] Among the many locations in Eau Claire, WI, where this research was done,the authors thank The Plus where we occasionally did research while enjoying theirpizza. We also thank the home of the third author Post’s uncles Rick and Mike whoseporch hosted our research team as we further investigated this work on Memorial Dayof 2017. Finally, we thank Phoenix Park where the first author Gullerud came up withthe proof of the spec containments.
4
2 Preliminaries and definitions
For a given n×n matrix A, a fundamental problem in linear algebra is the determination ofthe values λ for which the set of n homogeneous linear equations in n unknowns A~x = λ~x hasa non-trivial solution. The latter equation may be written in the form (A− λI)~x = ~0. Thegeneral theory of simultaneous linear algebraic equations shows that there is a non-trivialsolution if and only if the matrix A− λI is singular, that is, det(A− λI) = 0. This leads tothe following fundamental definitions used in this paper.
Definition 2.1 (Eigenvalue and eigenvector). Let A ∈ Cn×n be a square matrix. An eigen-pair of A is a pair (λ,~v) ∈ C× (Cn−~0) such that A~v = λ~v. We call λ an eigenvalue and itscorresponding nonzero vector ~v an eigenvector.
In particular, we often look at the set of eigenvalues of a given matrix A and the largestelement in absolute value of these eigenvalues defined as follows.
Definition 2.2 (Spectrum and spectral radius). Let A ∈ Cn×n be a square matrix. Themultiset of eigenvalues of A is called the spectrum of A and is denoted spec(A). The spectralradius of A is denoted ρ(A) and defined to be
ρ(A) = max{|λ| : λ ∈ spec(A)}.
The matrices we care about in this paper are a subset of a wider class of symmetricmatrices called Toeplitz matrices. A Toeplitz matrix is a matrix of the form
a0 a1 a2 . . . . . . an−1
a1 a0 a1. . .
...
a2 a1. . . . . . . . .
......
. . . . . . . . . a1 a2...
. . . a1 a0 a1an−1 . . . . . . a2 a1 a0
.
In particular, we focus on a subclass of Toeplitz matrices called tridiagonal symmetric ma-trices. We give the specific definitions below.
Definition 2.3 (Tridiagonal matrix). A tridiagonal symmetric matrix is a Toeplitz matrixin which all entries not lying on the diagonal or superdiagonal or subdiagonal are zero.
We limit our perspective by considering the tridiagonal matrices of the following form.
Definition 2.4 (The An matrices). Let An be an n × n tridiagonal symmetric matrix inwhich the diagonal entries are zero, and the superdiagonal and subdiagonal entries are allone.
5
Example 2.5. Here are the An matrices for n = 1, . . . , 4.
A1 =[0]
A2 =
[0 11 0
]A3 =
0 1 01 0 10 1 0
A4 =
0 1 0 01 0 1 00 1 0 10 0 1 0
A typical method of finding the eigenvalues of a square matrix is by calculating the roots
of what is called the characteristic polynomial of the matrix.
Definition 2.6 (The characteristic polynomial fn(λ)). Given the matrix An. The charac-teristic polynomial fn(λ) is given by
fn(λ) = det(An − λIn).
If we set fn(λ) = 0, then clearly we have that the n zeroes of this equation give thespectrum of An.
Example 2.7. Consider the following real symmetric matrix A4 and the correspondingdeterminant |A4 − λI4|.
A4 =
0 1 0 01 0 1 00 1 0 10 0 1 0
|A4 − λI4| =
∣∣∣∣∣∣∣∣−λ 1 0 01 −λ 1 00 1 −λ 10 0 1 −λ
∣∣∣∣∣∣∣∣ = λ4 − 3λ2 + 1.
Thus the characteristic polynomial is f4(λ) = λ4 − 3λ2 + 1. The roots of this polynomialyield the four distinct eigenvalues
λ1 =1 +√
5
2= φ ≈ 1.61803 λ3 =
−1 +√
5
2=
1
φ≈ .61803
λ2 =1−√
5
2≈ −.61803 λ4 =
−1−√
5
2≈ −1.61803,
where φ is the golden ratio. Moreover in Subsection 4.3, we determine an infinite set of nvalues for which these four eigenvalues in the n = 4 case are contained in the spectrum oflarger An matrices.
3 The characteristic polynomials of An
In this section we find the family of characteristic polynomials {fn(λ)}∞n=1 corresponding toour family of matrices. For each n, the coefficients of the polynomial fn(λ) reveal themselvesto be exactly the entries of the (n+1)th diagonal of Pascal’s triangle as denoted in Figure 3.1.We use this observation to find a closed form expression for fn(λ) for all n. To confirm thisexpression is indeed correct, we show that it satisfies the easily derived three-term recurrencerelation that the sequence {(fn(λ)}∞n=1 must follow.
6
3.1 A recurrence relation for fn(λ)
In this subsection, we show that the determinants of the matrices An − λIn, that is, thecharacteristic polynomials fn(λ), satisfy the following three-term recurrence relation
fn(λ) = −λfn−1(λ)− fn−2(λ)
with initial conditions f1(λ) = −λ and f2(λ) = λ2 − 1. First we consider the n × n matrixAn−λIn and note that bottom-right (n− 1)× (n− 1) submatrix is An−1−λIn−1, and hencethe minor given by deletion of the first row and first column is the characteristic polynomialfn−1(λ). This fact is pivotal in deriving the recurrence relation fn(λ).
An − λIn =
−λ 1 0 · · · 0
1 −λ 1. . . 0
0 1 −λ . . ....
.... . . . . . . . . 1
0 0 · · · 1 −λ
=
−λ 1 0 . . . 0
1
0 An−1−λIn−1...
0
Computing the determinant by cofactor expansion along the top row of An− λIn, we derivethe following.
fn(λ) = |An − λIn|
= −λ · |An−1 − λIn−1| − 1
∣∣∣∣∣∣∣∣∣∣∣∣
1 1 0 · · · 0
0 −λ 1. . . 0
0 1 −λ . . ....
.... . . . . . . . . 1
0 0 · · · 1 −λ
∣∣∣∣∣∣∣∣∣∣∣∣
= −λ · fn−1(λ)− 1
1 1 0 . . . 0
0... An−2−λIn−2...
0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣= −λ · fn−1(λ)− fn−2(λ).
3.2 The fn(λ) and their connection to Pascal’s triangle
If we compute the characteristic polynomials fn(λ) using the determinant of An − λIn, weget the following table of polynomials for n = 1, . . . , 12.
7
Figure 3.1: Pascal’s triangle
n-value fn(λ)
1 −1λ2 1λ2 −13 −1λ3 +2λ4 1λ4 −3λ2 +15 −1λ5 +4λ3 −3λ6 1λ6 −5λ4 +6λ2 −17 −1λ7 +6λ5 −10λ3 +4λ8 1λ8 −7λ6 +15λ4 −10λ2 +19 −1λ9 +8λ7 −21λ5 +20λ3 −5λ10 1λ10 −9λ8 +28λ6 −35λ4 +15λ2 −111 −1λ11 +10λ9 −36λ7 +56λ5 −35λ3 +6λ12 1λ12 −11λ10 +45λ8 −84λ6 +70λ4 −21λ2 +1
Upon examination of the coefficients of these twelve polynomials, it becomes immediatelyapparent that the binomial coefficients in the diagonal columns of Pascal’s triangle are inti-mately involved in the coefficients of the fn(λ). In Figure 3.1 we give the first twelve rowsof Pascal’s triangle, carefully coding the diagonal columns, to match the table of the twelvefn(λ) polynomials with their corresponding colors matching those in Pascal’s triangle.
8
3.3 A closed form fn(λ) for the recurrence relation
To arrive at our closed form for the solution to the recurrence relation, we use our observationof the intimate connection with Pascal’s triangle and the coefficients of the individual fn(λ)as n varies. For instance, the 1st, 2nd, etc. coefficients of each fn(λ) coincide directly withentries in the 1st, 2nd, etc.diagonals of Pascal’s triangle in the particular manner depicted inFigure 3.1. We highlight this connection to make this relationship apparent.
By careful analysis of the connection between the fn(λ) polynomials and Pascal’s triangle,we prove our main result that a closed form for the recurrence relation in Subsection 3.1 isgiven by
fm(λ) =
bm2 c∑i=0
(−1)m+i
(m− ii
)λm−2i. (1)
In actual practice, it is more helpful to consider the equation above for m even and m odd.The closed form when the index m is even is
f2k(λ) =k∑i=0
(−1)i(
2k − ii
)λ2k−2i. (2)
And the closed form when the index m is odd is
f2k+1(λ) =k∑i=0
(−1)i+1
((2k + 1)− i
i
)λ(2k+1)−2i. (3)
Before we prove that the recurrence relation fn(λ) = −λfn−1(λ)− fn−2(λ) is satisfied bythese closed forms above, we give a motivating example for when n is even. The case forwhen n is odd is similar.
Example 3.1. (An even case example) Set n = 10. According to the recurrence relation,
f10(λ) = −λf9(λ)− f8(λ). (?)
By Equation (2), the left hand side of (?) is
f10(λ) =5∑
i=0
(−1)i(
10− ii
)λ10−2i =
(10
0
)λ10 −
(9
1
)λ8 +
(8
2
)λ6 −
(7
3
)λ4 +
(6
4
)λ2 −
(5
5
).
By Equations (2) and (3), the right hand side of (?) is
−λf9(λ)− f8(λ) = −λ4∑
i=0
(−1)i+1
(9− ii
)λ9−2i −
4∑i=0
(−1)i(
8− ii
)λ8−2i
= −λ[−(90
)λ9 +
(81
)λ7 −
(72
)λ5 +
(63
)λ3 −
(54
)λ
]−
[(80
)λ8 −
(71
)λ6 +
(62
)λ4 −
(53
)λ2 +
(44
)]=
(90
)λ10 −
(81
)λ8 +
(72
)λ6 −
(63
)λ4 +
(54
)λ2 −
(80
)λ8 +
(71
)λ6 −
(62
)λ4 +
(53
)λ2 −
(44
)=
(90
)λ10 −
[(81
)+
(80
)]λ8 +
[(72
)+
(71
)]λ6 −
[(63
)+
(62
)]λ4 +
[(54
)+
(53
)]λ2 −
(44
)=
(100
)λ10 −
(91
)λ8 +
(82
)λ6 −
(73
)λ4 +
(64
)λ2 −
(55
)= f10(λ).
9
The second to last line above utilizes the fact that(90
)=(100
)and
(44
)=(55
)trivially, as well
as the property of Pascal’s triangle where(n
r
)+
(n
r + 1
)=
(n+ 1
r + 1
). (4)
We now state and prove our main result.
Theorem 3.2. Consider the recurrence relation fn(λ) = −λfn−1(λ) − fn−2(λ) with initialconditions f1(λ) = −λ and f2(λ) = λ2 − 1. A closed form for this recurrence relation isgiven by
fm(λ) =
bm2 c∑i=0
(−1)m+i
(m− ii
)λm−2i.
Proof. Consider the recurrence relation defined by
fn(λ) = −λfn−1(λ)− fn−2(λ).
It is readily verified that Equation (1) yields the initial conditions when n = 1, 2, so it sufficesto show that Equation (1) satisfies the recurrence relation. We divide this into two separatecases depending on the parity of n.CASE 1: (n is even)Since n ∈ 2N then n = 2k for some k ∈ N. Hence we want to show that the followingequation is satisfied by our appropriate choice of closed forms Equations (2) and (3) asneeded depending on the parity of each of the indices:
f2k(λ) = −λf2k−1(λ)− f2k−2(λ). (†)
We start with the right hand side of (†). For the ease of the reader, the expressions in redindicate the change from one line to the next.
−λf2k−1(λ)− f2k−2(λ)
= −λk−1∑i=0
(−1)i+1
((2k − 1)− i
i
)λ(2k−1)−2i −
k−1∑i=0
(−1)i(
(2k − 2)− ii
)λ(2k−2)−2i (5)
=
k−1∑i=0
(−1)i(
(2k − 1)− ii
)λ2k−2i +
k−1∑i=0
(−1)i+1
((2k − 2)− i
i
)λ(2k−2)−2i
=
(2k − 1
0
)λ2k +
k−1∑i=1
(−1)i(
(2k − 1)− ii
)λ2k−2i
+
k−2∑i=0
(−1)i+1
((2k − 2)− i
i
)λ(2k−2)−2i+ (−1)k
(k − 1
k − 1
)λ0
10
=
(2k − 1
0
)λ2k +
k−2∑j=0
(−1)j+1
((2k − 1)− (j + 1)
j + 1
)λ2k−2(j+1)
+
k−2∑j=0
(−1)j+1
((2k − 2)− j
j
)λ(2k−2)−2j + (−1)k
(k − 1
k − 1
)λ0
(6)
=
(2k − 1
0
)λ2k +
k−2∑j=0
(−1)j+1
[((2k − 1)− (j + 1)
j + 1
)+
((2k − 2)− j
j
)]λ2k−2−2j
+ (−1)k(k − 1
k − 1
)λ0
=
(2k − 1
0
)λ2k +
k−2∑j=0
(−1)j+1
((2k − 2− j) + 1
j + 1
)λ2k−2−2j + (−1)k
(k − 1
k − 1
)λ0 (7)
=
(2k
0
)λ2k +
k−1∑i=1
(−1)i(
2k − ii
)λ2k−2i + (−1)k
(2k − kk
)λ2k−2k (8)
=
k∑i=0
(−1)i(
2k − ii
)λ2k−2i
= f2k(λ) (9)
Line (6) holds by the closed forms given by Equations (2) and (3). Line (7) holds bytaking i = j + 1 for the first summation and i = j for the second summation. Line (8) holdsby Equation (4). Line (9) holds by taking i = j + 1 and realizing that
(2k−10
)=(2k0
)and(
k−1k−1
)=(kk
). Line (10) holds by the closed form given by Equation (2). Therefore the closed
forms satisfy the recurrence relation when n is even.
CASE 2: (n is odd)Since n ∈ 2N+ 1 then n = 2k+ 1 for some k ∈ N. Hence we want to show that the followingequation is satisfied by our appropriate choice of closed forms Equations (2) and (3) asneeded depending on the parity of each of the indices:
f2k+1(λ) = −λf2k(λ)− f2k−1(λ). (††)
We start with the right hand side of (††):
−λf2k(λ)− f2k−1(λ)
= −λk∑
i=0
(−1)i(
2k − ii
)λ2k−2i −
k−1∑i=0
(−1)i+1
((2k − 1)− i
i
)λ(2k−1)−2i (10)
=
k∑i=0
(−1)i+1
(2k − ii
)λ2k−2i+1 +
k−1∑i=0
(−1)i(
(2k − 1)− ii
)λ(2k−1)−2i
11
= −(
2k
0
)λ2k+1 +
k∑i=1
(−1)i+1
(2k − ii
)λ2k−2i+1
+
k∑j=1
(−1)j−1
((2k − 1)− (j − 1)
j − 1
)λ(2k−1)−2(j−1)
(11)
= −(
2k
0
)λ2k+1 +
k∑j=1
(−1)j+1
(2k − jj
)λ2k−2j+1 +
k∑j=1
(−1)j+1
(2k − jj − 1
)λ2k−2j+1 (12)
= −(
2k
0
)λ2k+1 +
k∑j=1
(−1)j+1
[(2k − jj
)+
(2k − jj − 1
)]λ2k−2j+1
= −(
2k + 1
0
)λ2k+1 +
k∑j=1
(−1)j+1
((2k + 1)− j
j
)λ(2k+1)−2j (13)
=
k∑j=0
(−1)j+1
((2k + 1)− j
j
)λ(2k+1)−2j
= f2k+1(λ) (14)
Line (12) holds by the closed forms given by Equations (2) and (3). Line (13) holdsby taking i = j − 1 for the second summation. Line (14) holds by taking i = j for thefirst summation. Line (15) holds by Equation (4) and realizing that
(2k0
)=(2k+10
). Line
(16) holds by the closed form given by Equation (3). Therefore the closed forms satisfy therecurrence relation when n is odd.
Hence Equation (1) satisfies the recurrence relation.
3.4 Parity properties of the fn(λ)
Observing the graphs of various characteristic functions fn(λ) in Figure 4.1, it appears thatthe functions of even index are symmetric about the y-axis (the hallmark trait of an evenfunction), while the functions of odd index are symmetric about the origin—that is, thegraph remains unchanged after 180 degree rotation about the origin (the hallmark trait ofan odd function). This observation prompts the following theorem.
Theorem 3.3. The characteristic equation fn(λ) is an even (respectively, odd) functionwhen n is even (respectively, odd).
Proof. It suffices to show
• Claim 1: f2k(−λ) = f2k(λ), and
• Claim 2: f2k+1(−λ) = −f2k+1(λ).
12
Using our closed form Equation (2) when the index is even, we see that Claim 1 holds since
(−λ)2k−2i =((−λ)2
)k−i= (λ2)k−i
= λ2k−2i.
(15)
And using our closed form Equation (3) when the index is odd, we see that Claim 2 holdssince
(−λ)(2k+1)−2i = (−λ)(−λ)2k−2i
= (−λ)λ2k−2i by the even case result (15)
= −(λ)(2k+1)−2i.
4 The spectrum of An
As mentioned in the introduction, the matrices An are the adjacency matrices of a veryimportant class of graphs, namely the path graphs. For example, below we give the pathgraph P4 and its corresponding adjacency matrix, which is our matrix A4.
v1 v2 v3 v4←→
v1 v2 v3 v4
v1 0 1 0 0v2 1 0 1 0v3 0 1 0 1v4 0 0 1 0
Hence these matrices are a fundamental object in spectral graph theory. Here is a table ofthe exact roots of the first five characteristic functions, and below that is a table of theseroots approximated up to 5 decimal places.
n-value exact roots of fn(λ)
1 λ = 02 λ = 1 or λ = −13 λ = 0 or λ =
√2 or λ = −
√2
4 λ = 12
(−√5− 1
)or λ = 1
2
(√5− 1
)or λ = 1
2
(1−√5)
or λ = 12
(√5 + 1
)5 λ = 0 or λ =
√3 or λ = −
√3 or λ = −1 or λ = 1
n-value roots (up to 5 decimal places) of fn(λ) in increasing order
1 λ = 02 λ = −1 or λ = 13 λ = −1.41421 or λ = 0 or λ = 1.414214 λ = −1.61803 or λ = −0.618034 or λ = 0.618034 or λ = 1.618035 λ = −1.73205 or λ = −1 or λ = 0 or λ = 1 or λ = 1.73205
13
In Figure 4.1, we plot the characteristic equations fn(λ) for n = 2, . . . , 7. Notice that sincethe odd-index equations have no constant term, the graphs of those equations necessarilypass through the origin. aBa needs to update this image to get rid of f6 and f7 plots
Figure 4.1: Plots of fn(λ) for n = 2, . . . , 7
Examining the roots in just these few small cases, we can visually confirm the following:
• As Theorem 3.3 guarantees, the polynomials f1, f3, and f5 are odd functions while thepolynomials f2 and f4 are even functions.
• If λ is a root of fn, then −λ is also a root.
• When n is even, all n roots are nonzero and come in distinct pairs λ and −λ.
• When n is odd, exactly one root is zero and the other n−1 roots come in distinct pairsλ and −λ.
• As n increases, the largest eigenvalue of An also increases and appears to be boundedabove by 2. More precisely, it seems that ρ(An) < 2 for all n.
In this section we explore in generality all the observations above. Regarding the spectrum ofthe family {An}∞n=1, we investigate the spectral radius, lower and upper bounds of spec(An),and a spectrum set-containment sufficiency condition for spec(Am) ⊂ spec(An) dependenton the values m,n ∈ N. We prove many of our assertions using a trigonometric closed formfor the roots of fn(λ)
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4.1 A trigonometric closed form for the roots of fn(λ)
Though many sources going back as far as 1965 are credited in later papers as giving closedforms for the eigenvalues of tridiagonal symmetric matrices, none of these cited journals andbooks actually provide a proof [1, 3, 6]. However in his 1978 book, Smith provides a proof forthe closed forms for eigenvalues of tridiagonal (but not necessarily) symmetric matrices thathave the values a, b, c ∈ C in the diagonal, superdiagonal, and subdiagonal, respectively [8].For these matrix values, he computes the n eigenvalues to be the formula
λs = a+ 2√bc cos
(sπ
n+ 1
)for s = 1, . . . , n.
Below we modify the proof to give a closed form expression for the eigenvalues of the Anmatrices.
Proposition 4.1. The n distinct eigenvalues of An are
λs = 2 cos
(sπ
n+ 1
)for s = 1, . . . , n.
Proof. Let λ represent an eigenvalue of An with corresponding eigenvector ~v. Consider theequation An~v = λ~v. This can be rewritten as (An − λIn)~v = ~0. Then
(An − λIn)~v =
−λ 1
1. . . . . .. . . . . . 1
1 −λ
~v1~v2...~vn
=
00...0
.The resulting system of equations is as follows:
−λ~v1 + ~v2 = 0
~v1 − λ~v2 + ~v3 = 0
......
~vj−1 − λ~vj + ~vj+1 = 0
......
~vn−2 − λ~vn−1 + ~vn = 0
~vn−1 − λ~vn = 0
In general, a single equation can be defined by
~vj−1 − λ~vj + ~vj+1 = 0 for j = 1, . . . , n (16)
where ~v0 = 0 and ~vn+1 = 0. Let a solution to this equation be ~vj = Amj for arbitrarynonzero constants A and m. Substituting this solution into Equation (16) shows that m isa root of
m2 − λm+ 1. (17)
15
Since Equation (17) is a quadratic equation, there are two, in our case real, roots. Hencethe general solution of Equation (16) is
~vj = Bmj1 + Cmj
2 (18)
where B and C are arbitrary nonzero constants and m1 and m2 are the two roots ofEquation (17). Since ~v0 = 0 and ~vn+1 = 0, by Equation (17), we get B + C = 0 andBmn+1
1 + Cmn+12 = 0, respectively. Substituting the former equation into the latter results
in (m1
m2
)n+1
= 1 = e2πis for s = 1, . . . , n.
Hence it follows thatm1
m2
= e2πis/(n+1). (19)
Since Equation (17) is a quadratic equation, the product of the roots is given by
m1m2 = 1. (20)
Then using Equation (19), it follows that
m1
m2
= e2πis/(n+1)
⇒ m1 = m2e2πis/(n+1)
⇒ m21 = m1m2e
2πis/(n+1)
⇒ m21 = e2πis/(n+1) by Equation (20)
⇒ m1 = eπis/(n+1).
Through a similar argument it can be found that
m2 = e−πis/(n+1).
Again since Equation (17) is a quadratic equation, the sum of the roots is given by
m1 +m2 = λ.
It follows that
λ = m1 +m2
= eπis/(n+1) + e−πis/(n+1)
= ei(πsn+1) + ei(
−πsn+1)
=
(cos
(πs
n+ 1
)+ i sin
(πs
n+ 1
))+
(cos
(−πsn+ 1
)+ i sin
(−πsn+ 1
))= 2 cos
(πs
n+ 1
)where the last equality holds since cosine is an even function and sine is an odd function.Thus we conclude λs = 2 cos
(πsn+1
)for s = 1, . . . , n.
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4.2 Spectral radius of An and some bounds questions
Much work has been done to study the bounds of the eigenvalues of real symmetric matrices.Methods discovered in the mid-nineteenth century reduce the original matrix to a tridiagonalmatrix whose eigenvalues are the same as those of the original matrix. Exploiting this idea,Golub in 1962 determined lower bounds on tridiagonal matrices of the form
a0 b1 0 . . . . . . 0
b1 a2 b2. . .
...
0 b2. . . . . . . . .
......
. . . . . . . . . bn−2 0...
. . . bn−2 an−1 bn−10 . . . . . . 0 bn−1 an
.
Proposition 4.2 (Golub [5, Corollary 1.1]). Let A be an n × n matrix with real entriesaij = ai for i = j, aij = bm for |i− j| = 1 where m = min(i, j), and aij = 0 otherwise. Thenthe interval [ak − σk, ak + σk] where σ2
k = b2k + b2k−1 contains at least one eigenvalue.
Using the proposition above, we can apply that to the tridiagonal matrices An. We easilyyield an interval in which the lower bound (in absolute value) of the eigenvalues of An isguaranteed to occur.
Corollary 4.3. For each matrix An, there exists an eigenvalue λ such that |λ| ≤ 1.
Proof. Given An, then in the language of Proposition 4.2, we have ai = 0 and bi = 1 for eachi. Thus the σk are calculated as follows
σk =
1, if k = 1√
2, if 2 ≤ k ≤ n− 1
1, if k = n
.
By Proposition 4.2, an eigenvalue is guaranteed to exist in the the interval [ak−σk, ak +σk].In particular for k = 1 the result follows.
So we have a firm interval where the lower bound (in absolute value) will contain aneigenvalue of An. As for an upper bound, it is clear by Proposition 4.1 that the followingtheorem holds.
Theorem 4.4. For each n ∈ N, the spectral radius ρ(An) of An is bounded above by 2.
4.3 Sufficient condition for spec(Am) ⊂ spec(An)
Recall in Section 3, we gave the characteristic equation when n = 7 as follows
f7(λ) = −λ7 + 6λ5 − 10λ3 + 4λ.
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For n = 7 the equation above has the following seven roots
λ = 0
λ = ±√
2
λ = ±√
2−√
2
λ = ±√√
2 + 2.
It is interesting to note that these exact seven roots appear as eigenvalues in a higher-degreecharacteristic equation. For example, when n = 15, the characteristic equation is
f15(λ) = −λ15 + 14λ13 − 78λ11 + 220λ9 − 330λ7 + 252λ5 − 84λ3 + 8λ.
The 15 distinct roots are the following
λ = 0
λ = ±√
2
λ = ±√
2−√
2
λ = ±√√
2 + 2
λ = ±√
2−√
2−√
2
λ = ±√√
2−√
2 + 2
λ = ±√
2−√√
2 + 2
λ = ±√√√
2 + 2 + 2.
This phenomenon is no coincidence. In fact the following theorem provides a way topredict the values m < n for which a complete set of eigenvalues of fm(λ) will be containedin the set of eigenvalues for fn(λ).
Theorem 4.5. Fix m ∈ N. For any n ∈ N such that m < n and n ≡ m (mod m + 1), itfollows that spec(Am) ⊂ spec(An).
Proof. Let m,n ∈ N such that m < n and n ≡ m (mod m+ 1). Then by definition
n = (m+ 1)k +m for some k ∈ N.
This can be rewritten asn = m(k + 1) + k. (21)
Now consider spec(Am) and spec(An). By Proposition 4.1, these are defined as follows:
spec(Am) = {λr | λr = 2 cos(
rπm+1
)}mr=1
spec(An) = {λs | λs = 2 cos(sπn+1
)}ns=1.
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We rewrite λs in terms of m and k:
λs = 2 cos
(sπ
n+ 1
)for s = 1, 2, . . . , n
= 2 cos
(sπ
m(k + 1) + k + 1
)for s = 1, 2, . . . ,m(k + 1) + k by Equation (21)
= 2 cos
(sπ
mk +m+ k + 1
)= 2 cos
(sπ
(m+ 1)(k + 1)
)= 2 cos
((s
k+1
)π
m+ 1
).
Recall λr = 2 cos(
rπm+1
). Hence λr = λs when r = s
k+1.
Since s ∈ S = {1, 2, . . . ,m(k + 1) + k}, the set T = {k + 1, 2(k + 1), . . . ,m(k + 1)} is asubset of S. Notice for each s ∈ T , there exists a unique r ∈ {1, 2, . . . ,m} under the equalityr = s
k+1. Thus for all λr ∈ spec(Am), there exists a λs ∈ spec(An) such that λr = λs. Then
λr ∈ spec(An) for all r ∈ {1, 2, . . . ,m}. Therefore spec(Am) ⊂ spec(An).
Remark 4.6. In the introduction we noted our observation that the golden ratio, its re-ciprocal, and their negatives arise as eigenvalues of An for the n values 4, 9, 14, 19, 24,29, 34, 39, and 44 via Mathematica. In Example 2.7, we computed the four eigenvaluesof A4. As an application of Theorem 4.5, if we let m = 4, then it is evident that thespec(A4) ⊆ spec(A4+5k) for all k ∈ N, thus confirming our observation.
5 Open questions
Question 5.1. In the plot in Figure 4.1 it appears that the maximum and minimum valuesfor the characteristic equations lie on a hyperbola. For example, here is a plot of f10(λ) andf20(λ).
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Do the max and min values of the sequence {fn(λ)}∞n=1 lie on a particular hyperbola? Andif so, can we find an exact formula for this hyperbola?
Question 5.2. A natural setting in which to generalize this research is the following familyof n× n matrices
0 b
b. . . . . .. . . . . . b
b 0
where b > 0 is some integer. These are the adjacency matrices of the multiple-edged pathgraphs, that is, the graphs that have b edges between each pair of consecutive vertices. Forexample, below we give the multiple-edge path graph on 4 vertices with two edges betweeneach vertex and its corresponding adjacency matrix.
v1 v2 v3 v4←→
v1 v2 v3 v4
v1 0 2 0 0v2 2 0 2 0v3 0 2 0 2v4 0 0 2 0
Do the coefficients of the corresponding characteristic polynomials have any connection tothe binomial coefficients as they do in the b = 1 case? Moreover, is there an analogue of thespectrum containment sufficiency condition as in our Theorem 4.5?
5.1 Conjectures on fn(λ) = Fn+1 where Fn+1 is the (n+1)th Fibonaccinumber
There is a tantalizing connection between the characteristic polynomials fn(λ) and the Fi-bonacci sequence {Fn}∞n=0 = {0, 1, 1, 2, 3, 5, 8, 13, 21, . . .}. Recall the polynomial f12(λ):
f12(λ) = 1λ12 − 11λ10 + 45λ8 − 84λ6 + 70λ4 − 21λ2 + 1.
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For what λ does f12(λ) = F13? It turns out λ = i is a root as follows:
f12(i) = 1i12 − 11i10 + 45i8 − 84i6 + 70i4 − 21i2 + 1
= 1 + 11 + 45 + 84 + 70 + 21 + 1
= 233 = F13.
We leave it to the reader to easily prove that we have the following result
Theorem 5.3. Let k ∈ N. Then λ = i satisfies f4k(λ) = F4k+1 where Fn denotes the nth
Fibonacci number.
However, something more compelling occurs when we graph the roots of the equationfn(λ) = Fn+1. For example, in the n = 12 case, we know that λ = i is a root. But whatabout the other 11 roots? Graphing them in the complex plane we get the following graphof the roots.
These roots appear to lie on an ellipse. This is not unique though to the fn where n is amultiple of 4. Consider the roots of f29(λ) = F30. This is the polynomial f29(λ):
f29(λ) = −λ29 + 28λ27 − 351λ25 + 2600λ23 − 12650λ21 + 42504λ19
− 100947λ17 + 170544λ15 − 203490λ13 + 167960λ11
− 92378λ9 + 31824λ7 − 6188λ5 + 560λ3 − 15λ.
Graphing the roots of f29(λ) = F30 in the complex plane, we obtain the following graph.
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What if we change a single coefficient? Change the coefficient of λ25 in f29(λ) from −351 to
−350 (call this modified polynomial f̃29(λ), and again set the polynomial to F30, we get
F30 = −λ29 + 28λ27−350λ25 + 2600λ23 − 12650λ21 + 42504λ19
− 100947λ17 + 170544λ15 − 203490λ13 + 167960λ11
− 92378λ9 + 31824λ7 − 6188λ5 + 560λ3 − 15λ.
Graphing the roots of f̃29(λ) = F30 in the complex plane, we obtain the following graph.
Observe that we no longer get an ellipse. Clearly there is something special about the rootsof fn(λ) = Fn+1. This leads on to the following conjecture.
Conjecture 5.4. The roots of fn(λ) = Fn+1 lie on an ellipse.
Lastly, from the two examples above of the graphs of the roots of f12(λ) = F13 andf29(λ) = F30, we observe that for the roots λ = a + bi, the imaginary parts b in absolutevalue appear to be bounded above by 1, while the real parts a in absolute value increasedslightly (in absolute value) from the value a ≈ 2.16648 in the n = 12 case to the valuea ≈ −2.20796 in the n = 29 case. Does this pattern continue to hold as we increase the nvalue in fn(λ) = Fn+1? Here are the roots of f201(λ) = F202 with a perfect ellipse fitting the201 distinct roots!
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In the above example, there is exactly one real root and it has value approximately −2.23206.Observe that this is larger in absolute value than the real root −2.20796 in the n = 29 case.Moreover, the root with the largest imaginary part is approximately (up to 10 decimal placesof accuracy) −0.0174265601 + 0.9999693615. Observe the imaginary part does not exceed1 but is larger than the imaginary part in the n = 29 setting. From further compellingevidence via Mathematica on a large number of concrete examples, we offer the followingconjecture.
Conjecture 5.5. Consider the roots a+bi of the equation fn(λ) = Fn+1. Then the followingholds:
• If n is even, then there are two distinct real roots and n − 2 distinct complex rootsoccurring in conjugate pairs.
• If n is odd, then there is one real roots with negative parity and n− 1 distinct complexroots occurring in conjugate pairs.
• The real parts a are unbounded as n increases, and
• The imaginary parts b are bounded above by 1 as n increases.
References
[1] M. Abromovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York(1965).
[2] I. Bar-On, Interlacing properties of tridiagonal symmetric matrices with applicationsto parallel computing, SIAM Journal on Matrix Analysis and Applications, 17 (1996),p. 548–562.
[3] S. Barnett, Matrices, Methods and Applications, Oxford University Press, Oxford (1990).
[4] C. Cryer, The numerical solution of boundary value problems for second order functionaldifferential equations by finite differences, Numerische Mathematik, 20 (1972), p. 288–299.
[5] G. Golub, Bounds for eigenvalues of tridiagonal symmetric matrices computed by theLR method, Mathematics of Computation 16 (1962), p. 428–447.
[6] W. Kahan and J. Varah, Two working algorithms for the eigenvalues of a symmetrictridiagonal matrix, Technical Report CS43, Computer Science Department, StanfordUniversity (1966), 29pp.
[7] L. Lu and W. Sun, The minimal eigenvalues of a class of block-tridiagonal matrices,IEEE Trans. Inform. Theory 43 (1997), p. 787–791.
[8] G.D. Smith, Numerical Solution of Partial Differential Equation: Finite Difference Meth-ods, 2nd ed., Clarendon Press, Oxford University Press, New York (1978).
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