Congruent Decomposition, Inertia, and Asymptotic Spectral Analysis of the Least Common Multiple (LCM) Matrix

This paper systematically studies the algebraic structure and asymptotic spectral properties of the least common multiple (LCM) matrix $A_n$ defined on the set $[n] = \{1, 2, \dots, n\}$. By employing the divisibility incidence matrix and Möbius inversion, we construct a framework for the nonsingular congruent diagonalization of the LCM matrix, namely $A_n = D_n E \Gamma E^T D_n$. Based on this decomposition and Sylvester's law of inertia, the inertia indices (the numbers of positive and negative eigenvalues) are precisely transformed into the problem of the parity distribution of the arithmetic function $\omega(k)$ (the number of distinct prime factors), and it is proved that the LCM matrix is always indefinite for $n \ge 2$. Combined with Halász's theorem from analytic number theory, it is rigorously proved that the positive and negative inertia indices tend to asymptotic equilibrium (each accounting for $1/2$) as $n \to \infty$. Through the Rayleigh quotient and estimates of sums over coprime integers, we establish that the largest eigenvalue is of order $\Theta(n^3)$. Moreover, using the algebraic structure of the inverse matrix and mean-value estimates for squarefree numbers, we prove that its $2$-norm condition number grows at most at a polynomial rate of $O(n^5 \log n)$, thereby ruling out the possibility of an exponential explosion of the condition number. The conclusions of this paper reveal the statistical symmetry of the spectral structure of the LCM matrix and its profound connection with the underlying distribution of prime numbers.