On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices

Jorma K. Merikoski and Pentti Haukkanen Faculty of Information Technology and Communication Sciences FI-33014 Tampere University Finland

                          jorma.merikoski@tuni.fi
                          pentti.haukkanen@tuni.fi

Antonio Sasaki Centre de Mathématiques Appliquées École Nationale Supérieure des Mines de

Paris Université Paris Sciences et Lettres FR-06560 Valbonne France

   antonio.sasaki@minesparis.psl.eu

Timo Tossavainen Department of Health, Education and Technology Luleå University of Technology SE-97187 Luleå Sweden

                                             timo.tossavainen@ltu.se

Abstract

We determine, for every n ≥ 1, the generalized eigenvalues of an n×n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an n×n LCM matrix to the corresponding GCD matrix when n ≤ 4, but breaks down for n > 4. In addition, we prove Cauchy’s interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence \underline{A004754} and the appearance of −1 as a generalized eigenvalue in the LCM–GCD setting.

1 Introduction

Let A and B be complex Hermitian n × n matrices, and suppose that B is positive definite; that is, the conjugate transpose A^{∗} and A are equal, and x^{∗}Bx > 0 whenever 0 ≠ x ∈ ℂ^{n}.

The generalized eigenvalue equation of A to B is

Ax = λBx, 0 ≠ x ∈ ℂ^{n}.  (1)

Then λ is called a generalized eigenvalue (“g-eigenvalue” for short) of A to B, and x is a corresponding generalized eigenvector (“g-eigenvector”). For more information, see, for example, Ghojogh, Karray, and Crowley [5]. They consider real symmetric matrices, yet all results extend naturally to complex Hermitian matrices.

It is actually enough that B is invertible in (1), and A can be arbitrary. However, the above assumptions are usually stated. Then all g-eigenvalues are real, and g-eigenvectors corresponding to distinct g-eigenvalues are orthogonal with respect to the inner product ⟨x, y⟩ = y^{∗}Bx.

The standard eigenvalues (“s-eigenvalues” for short) are widely studied. The g-eigenvalue equation (1) reduces to the s-eigenvalue equation

B^{−1}Ax = λx.

However, this “quick and dirty solution” [5] does not have significant use. So, g-eigenvalues must be considered in a different way. This area has not been studied much in the literature.

Let

S = {s_{1}, …, s_{n}}, s_{1} < ⋯ < s_{n},

be a set of positive real numbers. The n × n MAX matrix M_{S} and MIN matrix N_{S} on S are defined by

M_{S} = (m_{ij}^{S}), m_{ij}^{S} = max (s_{i}, s_{j}), N_{S} = (n_{ij}^{S}), n_{ij}^{S} = min (s_{i}, s_{j}).

Then N_{S} is positive definite [10, Theorem 8.1]. Also, let

T = {t_{1}, …, t_{n}}, t_{1} < ⋯ < t_{n},

be a set of positive integers. The n × n LCM matrix L_{T} and GCD matrix G_{T} on T are defined by

L_{T} = (l_{ij}^{T}), l_{ij}^{T} = lcm(t_{i}, t_{j}), G_{T} = (g_{ij}^{T}), g_{ij}^{T} = gcd (t_{i}, t_{j}).

Also, G_{T} is positive definite [3, Theorem 2].

We study g-eigenvalues of M_{S} to N_{S} in Section 2, and those of L_{T} to G_{T} in Sections 3 and 4. Finally, we complete our paper with discussion in Section 5.

All g-eigenvalues of A to A are trivially equal to one. We can therefore expect that, also in some nontrivial cases, the g-eigenvalues of A to B may be more accessible than the s-eigenvalues of A and B. We will see this in the case A = M_{S}, B = N_{S}, and also in the case A = L_{T}, B = G_{T}, where T = {1, …, n}, n ≤ 4. Recently, these matrices have been studied extensively (e.g., [1, 4, 6, 9, 10, 13]). These works discuss not only new results in this field but also applications to various other areas of mathematics—and, for instance, to computing [8], statistics [10], and signal processing [12].

2 MAX–MIN setting

We want to evaluate the g-eigenvalues of M_{S} to N_{S}, i.e., the solutions λ to the equation det(M_{S} − λN_{S}) = 0.

We begin with n = 2. Let S = {a, b}, 0 < a < b. Then

det (M_{S} − λN_{S}) = | \begin{matrix}a − λa & b − λa \\ b − λa & b − λb \end{matrix} | = a(b − a)λ^{2} − b(b − a) = 0

if and only if

λ = ±\sqrt{\frac{b}{a}}.

Our aim is to prove Theorem 1 below. However, because the general proof is not easily readable, we show the details only in the case n = 4. A careful reader will notice that we can proceed similarly for every integer n > 2.

Theorem 1. The g-eigenvalues of M_{S} to N_{S}, n > 2, are

λ_{1} = \sqrt{\frac{s_{n}}{s_{1}}} , λ_{2} = ⋯ = λ_{n−1} = −1, λ_{n} = −\sqrt{\frac{s_{n}}{s_{1}}} .  (2)

Proof. Let S = {a, b, c, d}, 0 < a < b < c < d. Then

M_{S} = ( \begin{matrix}a & b & c & d \\ b & b & c & d \\ c & c & c & d \\ d & d & d & d \end{matrix} ) , N_{S} = ( \begin{matrix}a & a & a & a \\ a & b & b & b \\ a & b & c & c \\ a & b & c & d \end{matrix} ) .

The matrix

M_{S} + N_{S} = ( \begin{matrix}2a & a + b & a + c & a + d \\ a + b & 2b & b + c & b + d \\ a + c & b + c & 2c & c + d \\ a + d & b + d & c + d & 2d \end{matrix} )

has rank 2 and nullity 2. Consequently, −1 is a g-eigenvalue of M_{S} to N_{S} with multiplicity 2. We show that the remaining g-eigenvalues are

λ = ±\sqrt{\frac{d}{a}}.

Regardless of the sign of λ, we have (note that d = λ^{2}a)

det (M_{S} + λN_{S}) = | \begin{matrix}a + λa & b + λa & c + λa & d + λa \\ b + λa & b + λb & c + λb & d + λb \\ c + λa & c + λb & c + λc & d + λc \\ d + λa & d + λb & d + λc & d + λd \end{matrix} |

= | \begin{matrix}a + λa & b + λa & c + λa & d + λa \\ b − a & λ(b − a) & λ(b − a) & λ(b − a) \\ c − b & c − b & λ(c − b) & λ(c − b) \\ d − c & d − c & d − c & λ(d − c) \end{matrix} | = | \begin{matrix}λa & b & c & λ^{2}a \\ b − a & λ(b − a) & λ(b − a) & λ(b − a) \\ c − b & c − b & λ(c − b) & λ(c − b) \\ d − c & d − c & d − c & λ(d − c) \end{matrix} |

+ | \begin{matrix}a & λa & λa & λa \\ b − a & λ(b − a) & λ(b − a) & λ(b − a) \\ c − b & c − b & λ(c − b) & λ(c − b) \\ d − c & d − c & d − c & λ(d − c) \end{matrix} | =: D_{1} + D_{2}.

Since D_{1} = D_{2} = 0, the claim follows. Remark 2. Theorem 1 holds also for n = 2. Then the chain λ_{2} = ⋯ = λ_{n−1} is “empty”. Remark 3. If the ordering of s_{1}, …, s_{n} is arbitrary, then (2) reads

λ_{1} = max_{i,j} \sqrt{\frac{s_{i}}{s_{j}}} , λ_{2} = ⋯ = λ_{n−1} = −1, λ_{n} = − max_{i,j} \sqrt{\frac{s_{i}}{s_{j}}} .  (3)

See also [10, Remark 2.1]. Remark 4. Theorem 1 applies also to the g-eigenvalues of M_{S} to N_{S′}, where

S′ = {s′_{1}, …, s′_{n}}, s′_{1} − s_{1} = ⋯ = s′_{n} − s_{n}.

3 LCM–GCD setting on T = {1, 2, …, n}

3.1 The case n ≤ 4

Let T = {1, 2, …, n}, let λ_{n1} ≥ ⋯ ≥ λ_{nn} be the g-eigenvalues of L_{T} to G_{T}, and let p_{n}(λ) = det (L_{T} − λG_{T}) be the g-characteristic polynomial. Then

p_{1}(λ) = 1 − λ, λ_{11} = 1,
p_{2}(λ) = | \begin{matrix}1 − λ & 2 − λ \\ 2 − λ & 2 − 2λ \end{matrix} | = λ^{2} − 2, λ_{21} = \sqrt{2}, λ_{22} = −\sqrt{2},
p_{3}(λ) = | \begin{matrix}1 − λ & 2 − λ & 3 − λ \\ 2 − λ & 2 − 2λ & 6 − λ \\ 3 − λ & 6 − λ & 3 − 3λ \end{matrix} | = −2(λ + 1)(λ^{2} − 6), λ_{31} = \sqrt{6}, λ_{32} = −1, λ_{33} = −\sqrt{6},

p_{4}(λ) = | \begin{matrix}1 − λ & 2 − λ & 3 − λ & 4 − λ \\ 2 − λ & 2 − 2λ & 6 − λ & 4 − 2λ \\ 3 − λ & 6 − λ & 3 − 3λ & 12 − λ \\ 4 − λ & 4 − 2λ & 12 − λ & 4 − 4λ \end{matrix} | = 4(λ + 1)^{2}(λ^{2} − 12),
λ_{41} = \sqrt{12}, λ_{42} = λ_{43} = −1, λ_{44} = −\sqrt{12}.

3.2 The case n > 4

The g-eigenvalues in Section 3.1 suggest that there may also be values of n > 4 such that

λ_{n1} = \sqrt{m}, λ_{n2} = ⋯ = λ_{n,n−1} = −1, λ_{nn} = −\sqrt{m}  (4)

for some integer m. We examine this hypothesis and begin with the case n = 5. We have

p_{5}(λ) = −16λ^{5} − 48λ^{4} + 528λ^{3} + 2480λ^{2} + 2880λ + 960
= −16(λ + 1)(λ^{4} + 2λ^{3} − 35λ^{2} − 120λ − 60)
=: −16(λ + 1)q(λ).

Because q(−1) = 24 ≠ 0, the multiplicity of λ = −1 is only one, falsifying (4). The geigenvalues are

λ_{51} = 6.4798, λ_{52} = −0.6118, λ_{53} = −1, λ_{54} = −3.3489, λ_{55} = −4.5191.

(These are approximations to four decimal places, similarly throughout the paper.) Interestingly, \sqrt{42} = 6.4807 is near to λ_{51}. If their difference were due to rounding errors, then the first equation in (4) would hold for n = 5, too. But p_{5}(\sqrt{42}) = −3168\sqrt{42} + 20448, showing that the difference is actual.

Moreover,

λ_{61} = 6.8501, λ_{62} = 2.5592, λ_{63} = −0.7419, λ_{64} = −1.3749, λ_{65} = −3.4396, λ_{66} = −5.8528.

Thus −1 is not a g-eigenvalue when n = 6.

These two cases already suffice to make it fairly clear that λ_{n2} = ⋯ = λ_{n,n−1} = −1 does not hold for n > 4. However, we choose to verify this claim thoroughly, as it involves first proving and then applying Cauchy’s interlacing theorem [7, Theorem 4.3.17] for geigenvalues—a result that is arguably of general interest. To that end, recall that there are two positive g-eigenvalues for n = 6. By Theorem 6 below, there are at least two positive g-eigenvalues for n = 7. Continuing in this way confirms the claim.

We conclude this section by exploring in which dimensions −1 occurs as a g-eigenvalue. Computer experiments covering the range 1 ≤ n ≤ 1000 (with code provided in the Appendix) show that −1 is a g-eigenvalue if and only if

n = \overbrace{4, 5}^{2}, \overbrace{8, 9, 10, 11}^{2^{2}}, \overbrace{16, …, 23}^{2^{3}}, \overbrace{32, …, 47}^{2^{4}}, \overbrace{64, …, 95}^{2^{5}}, \overbrace{128, …, 191}^{2^{6}}, \overbrace{256, …, 383}^{2^{7}}, 512, …,

where the overbrace indicates the number of terms. This sequence is the same as the OEIS [11] sequence \underline{A004754} without the first term. Its description [11] raises an interesting conjecture.

Conjecture 5. Let T = {1, …, n}, n > 3. Then −1 is a g-eigenvalue of L_{T} to G_{T} if and only if the binary representation of n begins with 10.

For example, 4 = (100)_{2}, 5 = (101)_{2}, 8 = (1000)_{2}, 19 = (10011)_{2}.

This OEIS sequence (a_{n}) satisfies [11]

a_{2^{m}+k} = 2^{m+1} + k, m ≥ 0, 0 ≤ k < 2^{m}.  (5)

If, for example, m = k = 3, then the left-hand side equals a_{8+3} = a_{11} = 19, and the righthand side equals 16 + 3 = 19. An induction proof of Conjecture 5 can perhaps be found by using (5).

3.3 Cauchy’s interlacing theorem for g-eigenvalues

Theorem 6. Let A and B be as in (1), n > 1, with first leading principal submatrices A′ and respectively B′ (obtained by removing the nth row and column). Let

λ_{1} ≥ ⋯ ≥ λ_{n} and λ′_{1} ≥ ⋯ ≥ λ′_{n−1}

be the g-eigenvalues of A to B and, respectively, of A′ to B′. Then

λ_{1} ≥ λ′_{1} ≥ λ_{2} ≥ λ′_{2} ≥ ⋯ ≥ λ_{n−1} ≥ λ′_{n−1} ≥ λ_{n}.

Proof. Let ⪯ denote the subspace inclusion. Because the Courant-Fischer theorem [7, Theorem 4.2.6] extends to g-eigenvalues [2, Theorem 3] (note the wrong ordering of max and min in its formulation), we have

λ_{k} = max_{\begin{matrix}U⪯ℂ^{n} \\ dim U=k \end{matrix}} min_{0≠x∈U} \frac{x^{∗}Ax}{x^{∗}Bx}, k = 1, …, n,  (6)

and

λ′_{k} = max_{\begin{matrix}V ⪯ℂ^{n−1} \\ dim V =k \end{matrix}} min_{0≠y∈V} \frac{y^{∗}A′y}{y^{∗}B′y}, k = 1, …, n − 1.  (7)

Let

0 ≠ x ∈ ℂ^{n}, x = ( \begin{matrix}x′ \\ x_{n} \end{matrix} ) , A = ( \begin{matrix}A′ & u \\ u^{∗} & a_{nn} \end{matrix} ) , B = ( \begin{matrix}B′ & v \\ v^{∗} & b_{nn} \end{matrix} ) .

If x_{n} = 0, then

\frac{x^{∗}Ax}{x^{∗}Bx} = \frac{(x′)^{∗}A′x′}{(x′)^{∗}B′x′} ,

so, for k = 1, …, n − 1,

λ′_{k} = max_{\begin{matrix}V ⪯ℂ^{n−1} \\ dim V =k \end{matrix}} min_{\begin{matrix}0≠x′∈V \\ x_{n}=0 \end{matrix}} \frac{x^{∗}Ax}{x^{∗}Bx} =: M.

Since

\{ min_{\begin{matrix}0≠x′∈V \\ x_{n}=0 \end{matrix}} \frac{x^{∗}Ax}{x^{∗}Bx} : V ⪯ ℂ^{n−1}, dim V = k\} ⊆ \{ min_{0≠x∈U} \frac{x^{∗}Ax}{x^{∗}Bx} : U ⪯ ℂ^{n}, dim U = k\},

it follows that

M ≤ max_{\begin{matrix}U⪯ℂ^{n} \\ dim U=k \end{matrix}} min_{0≠x∈U} \frac{x^{∗}Ax}{x^{∗}Bx}.  (8)

Now, by (7), (8), and (6),

λ′_{k} ≤ λ_{k}.

To find a reverse inequality, we change the ordering of max and min in the generalized Courant-Fischer theorem:

λ_{k+1} = min_{\begin{matrix}U⪯ℂ^{n} \\ dim U=n−k \end{matrix}} max_{0≠x∈U} \frac{x^{∗}Ax}{x^{∗}Bx}, k = 0, …, n − 1,

and

λ′_{k} = min_{\begin{matrix}V ⪯ℂ^{n−1} \\ dim V =n−k \end{matrix}} max_{0≠y∈V} \frac{y^{∗}A′y}{y^{∗}B′y}, k = 1, …, n − 1.

By a simple modification of the previous argument, we obtain

λ′_{k} ≥ λ_{k+1},

completing the proof.

4 LCM–GCD setting on some T ≠ {1, 2, …, n}

4.1 The cases n = 2, 3

First, let T = {u, v}, 0 < u < v. Studying {u/d, v/d} if d = gcd (u, v) > 1, we can assume that gcd (u, v) = 1. Then

det (L_{T} − λG_{T}) = | \begin{matrix}u − λu & uv − λ \\ uv − λ & v − λv \end{matrix} | = (uv − 1)(λ^{2} − uv), λ_{1} = \sqrt{uv}, λ_{2} = −\sqrt{uv}.

Next, let T = {1, u, v}, where 1 < u < v and gcd (u, v) = 1. Then

det (L_{T} − λG_{T}) = | \begin{matrix}1 − λ & u − λ & v − λ \\ u − λ & u − λu & uv − λ \\ v − λ & uv − λ & v − λv \end{matrix} |

= (u + v − uv − 1)λ^{3} + (u + v − uv − 1)λ^{2} + (u^{2}v^{2} + uv − u^{2}v − uv^{2})λ + u^{2}v^{2} − u^{2}v − uv^{2} + uv = (u − 1)(v − 1)(λ + 1)(λ^{2} − uv), λ_{1} = \sqrt{uv}, λ_{2} = −1, λ_{3} = −\sqrt{uv}.

More generally, let T = {u, v, w}, where 1 < u < v < w and gcd (u, v) = gcd (u, w) = gcd (v, w) = 1. It seems that we do not get pretty results. If, for example, T = {2, 3, 5}, then

det (L_{T} − λG_{T}) = | \begin{matrix}2 − 2λ & 6 − λ & 10 − λ \\ 6 − λ & 3 − 3λ & 15 − λ \\ 10 − λ & 15 − λ & 5 − 5λ \end{matrix} | = −22λ^{3} − 38λ^{2} + 420λ + 900,

λ_{1} = 4.5128, λ_{2} = −2.3027, λ_{3} = −3.9371.

4.2 The case T = {1, p, …, p^{n−1}}, p ∈ ℙ

In this case,

L_{T} = ( \begin{matrix}1 & p & p^{2} & ⋯ & p^{n−1} \\ p & p & p^{2} & ⋯ & p^{n−1} \\ p^{2} & p^{2} & p^{2} & ⋯ & p^{n−1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ p^{n−1} & p^{n−1} & p^{n−1} & ⋯ & p^{n−1} \end{matrix} ) = M_{T} , G_{T} = ( \begin{matrix}1 & 1 & 1 & ⋯ & 1 \\ 1 & p & p & ⋯ & p \\ 1 & p & p^{2} & ⋯ & p^{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & p & p^{2} & ⋯ & p^{n−1} \end{matrix} ) = N_{T} .

By Theorem 1, the g-eigenvalues of L_{T} to G_{T} are

λ_{1} = p^{\frac{n−1}{2}} , λ_{2} = ⋯ = λ_{n−1} = −1, λ_{n} = −p^{\frac{n−1}{2}} .

4.3 Reordering does not matter

We noted in Remark 3 that reordering S in the MAX–MIN setting only changes (2) to (3), so all g-eigenvalues remain unchanged. We now show that reordering T in the LCM–GCD setting also preserves the g-eigenvalues. More generally, let X = (x_{ij}) be a complex square matrix of order n. Given a permutation σ of (1, …, n), define

X_{σ} = (x_{ij}^{σ}), x_{ij}^{σ} = x_{σ(i),σ(j)},

and let P_{σ} denote the permutation matrix corresponding to σ. Since

X_{σ} = P_{σ}XP_{σ} and det P_{σ} = ±1,

we have

det X_{σ} = det X,

implying the claim.

5 Discussion

Above, we first examined the g-eigenvalues of MAX matrices to MIN matrices. The results reveal distinct structural patterns: the g-eigenvalues are completely characterized and consist of one positive value, several values −1 (with multiplicity zero when n = 2), and one negative value. This regularity highlights an underlying symmetry and robustness in the generalized eigenstructure of these matrices.

We then turned to the g-eigenvalues of LCM matrices to GCD matrices on T = {1, …, n}. This case is more intricate. While the above pattern holds for n ≤ 4, it breaks down for n > 4. In the course of verifying this, we proved a generalization of Cauchy’s interlacing theorem for eigenvalues—namely, the corresponding theorem for g-eigenvalues.

A surprising observation is the emergence of connection to OEIS sequence \underline{A004754} in Conjecture 5. If proven, this would provide a novel bridge between matrix theory and number theory, offering a new insight into exploring spectral properties of matrices through binary representations of integers.

We concluded our study by considering sets T ≠ {1, …, n} to demonstrate that certain configurations in the LCM–GCD setting exhibit a MAX–MIN structure. We also showed that, in general, reordering S and T does not affect the g-eigenvalues, thereby reinforcing the robustness of these matrices under permutations.

From a computational perspective, determining generalized eigenvalues poses significant challenges, as it typically requires finding the roots of high-degree characteristic polynomials. As n increases, these polynomials become difficult to construct and numerically unstable to solve. However, in constructing the sequence in Conjecture 5, these difficulties can largely be avoided: it is not necessary to form or factorize p_{n}(λ) = det(L_{T} − λG_{T}) explicitly. Instead, one can directly compute p_{n}(−1) = det(L_{T} +G_{T}). This approach is computationally lighter, numerically more stable, and sufficient to verify whether −1 is a g-eigenvalue.

Appendix

1 import  numpy  as np
2 import  math
3 from  scipy.linalg  import  eig

4

5 def  gcd_matrix(n):
6  """Construct  an n x n matrix  with  entries  gcd(i, j)."""
7  M = np.zeros((n, n), dtype=int)
8  for i in  range(1, n+1):
9  for j in  range(1, n+1):
10  M[i-1, j-1] = math.gcd(i, j)
11  return M

12

13 def  lcm_matrix(n):
14  """Construct  an n x n matrix  with  entries  lcm(i, j)."""

15  M = np.zeros((n, n), dtype=int)
16  for i in  range(1, n+1):
17  for j in  range(1, n+1):
18  M[i-1, j-1] = math.lcm(i, j)
19  return M

20

21 def  find_n_with_minus_one(tol=1e-5, max_n=1000):
22  """
23  For n = 1 to max_n, computes  the  generalized  eigenvalues  for Ax =
           lambda Bx, where A is the  LCM  matrix  and B is the  GCD  matrix.
           Returns a list of n for  which  -1 appears  as a generalized
           eigenvalue (within a tolerance  tol).
24  """
25  n_list = []
26  for n in  range(1, max_n+1):
27  A = lcm_matrix(n).astype(float)
28  B = gcd_matrix(n).astype(float)
29  # Compute g-eigenvalues  of A to B:
30  eigenvalues, _= eig(A, B)
31  # Convert  real g-eigenvalues  to real  values:
32  eigenvalues = np.real_if_close(eigenvalues, tol=tol)
33  # Check  if some g-eigenvalue  is  equal  to  -1:
34  if any(np.isclose(ev,  -1, atol=tol) for ev in  eigenvalues):
35  n_list.append(n)
36  return  n_list

37

38 if  __name__== '__main__':
39  result = find_n_with_minus_one(tol=1e-5, max_n=1000)
40  print("Dimensions n for  which  -1 appears  as a g-eigenvalue:")
41  print(result)

Listing 1: Python code to examine Conjecture 5.

References

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2020 Mathematics Subject Classification: Primary 15A18; Secondary 11C20. Keywords: generalized eigenvalue, Cauchy’s interlacing theorem, Hermitian matrix, GCD matrix, LCM matrix, MIN matrix, MAX matrix.

(Concerned with sequences \underline{A001088}, \underline{A003983}, \underline{A004754}, \underline{A051125}, and \underline{A060238}.)

Received May 5 2025; revised version received November 18 2025. Published in Journal of Integer Sequences, December 2 2025. Return to Journal of Integer Sequences home page.