On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]


               <jats:p>Given a real number <jats:italic>a</jats:italic> ≥ 1, let <jats:italic>K<jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>a</jats:italic>) be the set of all <jats:italic>n</jats:italic> × <jats:italic>n</jats:italic> unit lower triangular matrices with each element in the interval [−<jats:italic>a</jats:italic>, <jats:italic>a</jats:italic>]. Denoting by <jats:italic>λ<jats:sub>n</jats:sub>
                  </jats:italic>(·) the smallest eigenvalue of a given matrix, let <jats:italic>c<jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>a</jats:italic>) = min {λ <jats:italic>
                     <jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>YY<jats:sup>T</jats:sup>
                  </jats:italic>) : <jats:italic>Y</jats:italic> ∈ <jats:italic>K<jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>a</jats:italic>)}. Then <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2020-0139_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
                           <m:mrow>
                              <m:msqrt>
                                 <m:mrow>
                                    <m:msub>
                                       <m:mrow>
                                          <m:mi>c</m:mi>
                                       </m:mrow>
                                       <m:mi>n</m:mi>
                                    </m:msub>
                                    <m:mrow>
                                       <m:mo>(</m:mo>
                                       <m:mi>a</m:mi>
                                       <m:mo>)</m:mo>
                                    </m:mrow>
                                 </m:mrow>
                              </m:msqrt>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>\sqrt {{c_n}\left( a \right)}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is the smallest singular value in <jats:italic>K<jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>a</jats:italic>). We find all minimizing matrices. Moreover, we study the asymptotic behavior of <jats:italic>c<jats:sub>n</jats:sub>
                  </jats:italic>(<jats:italic>a</jats:italic>) as <jats:italic>n</jats:italic> → ∞. Finally, replacing [−<jats:italic>a</jats:italic>, <jats:italic>a</jats:italic>] with [<jats:italic>a</jats:italic>, <jats:italic>b</jats:italic>], <jats:italic>a</jats:italic> ≤ 0 < <jats:italic>b</jats:italic>, we present an open question: Can our results be generalized in this extension?</jats:p>


Introduction
Let S = {x , x , . . . , xn} be a set of distinct positive integers and let (x i , x j ) and [x i , x j ] denote the greatest common divisor and the least common multiply of x i and x j , respectively. The n×n matrices (S) = ((x i , x j )) and [S] = ([x i , x j ]) are called the GCD matrix and the LCM matrix on S, respectively. Many results on these matrices, their various generalizations and relatives have been published in the literature. For general accounts, see [2-4, 6, 8, 12] and the references therein.
One of the richest topics in the study of GCD and LCM matrices is their spectral properties. In this frame, Hong and Loewy [6] studied the asymptotic behavior of the eigenvalues of power GCD matrices and introduced a new parameter to present a lower bound for the smallest eigenvalue of the power GCD matrix. They de ned the numbers cn depending only on n as follows: where λn(·) is the smallest eigenvalue and Kn is the set of all n×n nonsingular lower triangular ( , )-matrices. Then, in the light of their MATLAB calculations for n = , , . . . , , Ilmonen, Haukkanen and Merikoski [8] Verifying the truth of the conjecture for n = and in [1], the author of the present paper, Keskin, Yıldız and Demirbüken [4] proved the conjecture and realized that there is only one matrix Y ∈ Kn for which cn is attained. Therefore, it was conjectured that if cn = λn(YY T ) for Y ∈ Kn, then Y = Y [4, Conjecture 3.1]. Recently, Loewy [11] has proved the conjecture.
Due to the importance of the study of spectral properties of GCD and related matrices, some authors tried to nd bounds for cn. In this frame, assuming the truth of the conjecture, Mattila [12] obtained the following lower bound cn ≥ n + n + n n− for even n, cn ≥ n + n + n − n− for odd n.
Beside this, Merikoski, who contributed to the paper [4], improved Mattila's lower bound as follows: where Fn is the nth Fibonacci number. In addition to these bounds, in [2] the inequality where µ is the Möbius function, was proposed for cn by using a di erent method. Kaarnioja [9] improved the lower bound further and showed that is the golden ratio. He also conjectured asymptotically that cn ∼ φ n , which was proved by Loewy [11]. This paper has two main goals. The rst is to expand the existing results on the process of nding all matrices for which √ cn is attained. The second is to determine the asymptotic behaviour of cn in a larger class of matrices. Given a real number a ≥ , let Kn(a) denote the set of all n × n unit lower triangular matrices whose each element under diagonal is in the interval [−a, a]. We note that all diagonal entries of a unit lower triangular matrix are 1. Let where λn(·) is the smallest eigenvalue. It is clear that cn(a) is the smallest singular value in Kn(a). Let Y = (y ij ) ∈ Kn(a) be de ned by for i > j. In Section 2, we present a sharp upper bound for the absolute values of entries of Y − and (YY T ) − , where Y ∈ Kn(a), and obtain all the entries of Y − and (Y Y T ) − in terms of a. In Section 3, we prove that cn(a) = λn(Y Y T ). We also show that if cn(a) = λn( Moreover, we determine the asymptotic behavior of cn(a). In other words, we show that cn(a) ∼ (a+ ) (a+ ) n − as n → ∞. Finally, we present concluding remarks in Section 4, including some open problems related to a certain generalization of the results on cn and cn(a).

Preliminaries
In order to nd the bound mentioned in the introduction, we present ve lemmas here. In the rst lemma, we obtain a recurrence relation for the entries of inverses of matrices in Kn(a).
Proof. See [4, Lemma 2.2]. The set of matrices is di erent, but the proof works also here.

Lemma 2. [5] Let Y and Y − be as in Lemma 1. Then
Now assume that the inequality b j+t,j ≤ a(a + ) t− holds for each t = , , . . . , k − . We show that b j+k,j ≤ a(a + ) k− . By Lemma 1 and the induction hypothesis, we have The induction principle completes the proof.

Lemma 3. [5]
Let Y be as in (2) and Y − = (c ij ). Then Since Y ∈ Kn(a), we have by Lemma 1 Next, we show by induction on t = i − j that If t = , then c j+ ,j = −w j+ ,j = −a.
Now assume that (3)  The induction principle completes the proof.
In the following, we denote by (·) ij the ij-th entry of a given matrix. (2) and Z = Y Y T . Then,

Lemma 4. [5] Let Y be as in
Proof. By Lemma 3, we have for ≤ i < j ≤ n. Since Z is symmetric, the claim follows.

Lemma 5. [5] Let Z = YY T , Y ∈ Kn(a), and let Z be as in Lemma 4. Then |Z
Proof. Let Y − be as in Lemma = (Z − ) ij .

Main Results
Theorem 1. Let Y be as in (2). Then cn(a) = λn(Y Y T ). Conversely, if Y ∈ Kn(a) and λn(YY T ) = cn(a), then Proof. For the rst claim, we proceed as in the proof [4] of the Ilmonen-Haukkanen-Merikoski conjecture.
Let Z = Y Y T . Then, by (4) and (5), trZ −k = tr|Z − | k for all positive integers k. By Newton's identities [10], one can easily see that Z − and |Z − | have the same characteristic polynomial. Since they are real symmetric matrices, their spectral radii are equal, that is, ρ(|Z − |) = ρ(Z − ). Let Y ∈ Kn(a) and Z = YY T . By Lemma 5 and [7, Theorem 8.1.18], we obtain the following inequalities Since Z − and Z − are positive de nite, we have where λ (·) is the largest eigenvalue. Hence, by (8), Let Y − = (b ij ) and Y − = (c ij ). Since Y − is a lower triangular matrix, inequalities (6) and (7), and equality (9) together imply that ⇐⇒ ϵ j,i = ϵ j+ ,i ϵ j+ ,j = ϵ j+ ,i ϵ j+ ,j = · · · = ϵ n,i ϵ n,j for ≤ i < j ≤ n. This means that every ϵ j,i depends only ϵ n,i and ϵ n,j . Therefore, setting ε i = ϵ n,i for all i = , , . . . , n, for the sake of brevity, we have (11) Proof. To prove the theorem, we use a similar but not the same method to those of Theorem 2.1 in [11]. by the induction hypothesis and the assumption a ≥ . Thus, we have det(H n+ ) < det(Hn) < for all n > . Next, we show that every eigenvalue of Z − , except its spectral radius ρ(Z − ), lies in the interval [ , ]. Let λ (n) ≥ λ (n) ≥ · · · ≥ λ (n) n be the eigenvalues of the n × n matrix Z − . It is clear that λ (n) n > as Z − is positive de nite. We show again by induction on n that λ (n) > and λ (n) < . For n = , λ (n) = a + + (a + ) − > and λ (n) = a + − (a + ) − < .