Cauchy's Interlacing Theorem for Generalized Eigenvalues

• Cauchy's interlacing theorem for generalized eigenvalues defines how the spectra of principal subpencils of Hermitian matrix pairs interlace with those of the full pencil.
• It employs variational and minimax characterizations to establish precise eigenvalue bounds, enhancing spectral estimates and stability analyses.
• The theorem informs computational strategies for arithmetic matrices and supports applications in vibration analysis, constrained optimization, and numerical linear algebra.

Cauchy's Interlacing Theorem for Generalized Eigenvalues provides a precise description of how the spectra of principal subpencils of Hermitian matrix pairs relate to the spectra of the full pencil when considering the generalized eigenvalue problem. Given Hermitian $A, B \in \mathbb{C}^{n \times n}$ with $B$ invertible, a generalized eigenvalue (g-eigenvalue) $\lambda \in \mathbb{R}$ satisfies $A x = \lambda B x$ for some $x \neq 0$. When $B$ is positive definite (or, more generally, invertible), all such eigenvalues are real and can be ordered nondecreasingly. The interlacing theorem extends the classical Cauchy interlacing phenomenon to the pencil $(A, B)$, asserting that the g-eigenvalues of principal subpencils interlace those of the full pencil. This property underlies spectral estimates and inductive spectral arguments and has further implications for the structure and computation of generalized eigenvalues, especially in the context of arithmetic matrices such as MAX, MIN, LCM, and GCD matrices (Merikoski et al., 23 Jan 2026).

1. Generalized Eigenvalue Problem and Notation

For Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$, with $B$ invertible, the generalized eigenvalue problem asks for $\lambda \in \mathbb{R}$ and $x \neq 0$ such that

$A x = \lambda B x.$

If $B$ is Hermitian and invertible, all generalized eigenvalues are real and are typically ordered as

$$\lambda_1(A, B) \leq \lambda_2(A, B) \leq \cdots \leq \lambda_n(A, B).$$

For any index set $I \subset \{1, \ldots, n\}$ with $|I| = k$, the principal subpencils $A[I], B[I] \in \mathbb{C}^{k \times k}$ are obtained by deleting all rows and columns outside $I$. Their g-eigenvalues are denoted by

$$\lambda_1(A[I], B[I]) \leq \cdots \leq \lambda_k(A[I], B[I]).$$

2. Statement of the Interlacing Theorem

Cauchy's interlacing theorem for generalized eigenvalues states that for every principal subpencil determined by $I \subset \{1, \ldots, n\}$ with $|I| = k$, and for every $i = 1, \ldots, k$,

$$\lambda_i(A[I], B[I]) \leq \lambda_i(A, B) \leq \lambda_{i + (n-k)}(A[I], B[I]).$$

In particular, for the leading principal subpencil $(A', B')$ of size $(n-1) \times (n-1)$, the interlacing simplifies to

$$\lambda_i(A, B) \leq \lambda_i(A', B') \leq \lambda_{i+1}(A, B), \quad i=1,\ldots,n-1.$$

3. Variational and Minimax Characterization

The generalized Courant–Fischer theorem provides a variational description of the g-eigenvalues:

$$\lambda_k(A, B) = \max_{U \subset \mathbb{C}^n,~\dim U = k}~\min_{0 \neq x \in U} \frac{x^* A x}{x^* B x}, \quad k=1, \ldots, n,$$

with a dual form:

$$\lambda_k(A, B) = \min_{V \subset \mathbb{C}^n,~\mathrm{codim}\, V = n-k+1}~\max_{0 \neq x \in V} \frac{x^* A x}{x^* B x}.$$

To prove the interlacing, principal subspaces of $\mathbb{C}^{n-1}$ can be embedded in $\mathbb{C}^n$ by extending vectors with a zero in the last coordinate. In the maximization over $k$-dimensional subspaces, the restriction imposed by working in the principal subspace yields

$\lambda_k(A', B') \leq \lambda_k(A, B)$

("first half" of interlacing), while the dual minimization argument provides the second inequality

$\lambda_k(A, B) \leq \lambda_{k+1}(A', B').$

Combined, these yield the full interlacing. The same approach with appropriate embeddings proves the general $I$-indexed form.

4. Explicit Example: MAX–MIN Matrix Pencil

Consider $S = \{1, 2, 3\}$ with

$$M = [\max(i,j)]_{i,j=1}^3 = \begin{bmatrix} 1 & 2 & 3 \ 2 & 2 & 3 \ 3 & 3 & 3 \end{bmatrix}, \quad N = [\min(i,j)]_{i,j=1}^3 = \begin{bmatrix} 1 & 1 & 1 \ 1 & 2 & 2 \ 1 & 2 & 3 \end{bmatrix}.$$

The g-eigenvalues of $(M, N)$ are:

$$\lambda_1(M, N) = -\sqrt{3} \approx -1.732, \quad \lambda_2 = -1, \quad \lambda_3 = +\sqrt{3} \approx +1.732.$$

For the leading $2 \times 2$ subpencil on indices $\{1,2\}$,

$$M' = \begin{bmatrix} 1 & 2 \ 2 & 2 \end{bmatrix}, \quad N' = \begin{bmatrix} 1 & 1 \ 1 & 2 \end{bmatrix}$$

with g-eigenvalues

$$\lambda_1(M', N') = -\sqrt{2} \approx -1.414, \quad \lambda_2 = +\sqrt{2} \approx +1.414.$$

The interlacing can be verified concretely: | Index $i$ | $\lambda_i(M, N)$ | $\lambda_i(M', N')$ | $\lambda_{i+1}(M, N)$ | |:---------:|:----------------:|:------------------:|:---------------------:| | 1 | $-1.732$ | $-1.414$ | $-1$ | | 2 | $-1$ | $+1.414$ | $+1.732$ |

The inequalities

$$\lambda_1(M, N) \leq \lambda_1(M', N') \leq \lambda_2(M, N), \quad \lambda_2(M, N) \leq \lambda_2(M', N') \leq \lambda_3(M, N)$$

are satisfied.

5. Nonstandard Phenomena, Conjectures, and Computational Aspects

For the pencil $(\mathrm{LCM}_n, \mathrm{GCD}_n)$, a notable pattern is observed for $n \leq 4$: one large positive eigenvalue, multiple $-1$s, and one large negative eigenvalue. For $n > 4$ this pattern "breaks down." Repeated application of interlacing demonstrates that once $-1$ loses its high multiplicity, it cannot reacquire it at larger $n$. The authors conjecture, based on OEIS A004754, that $-1$ is a generalized eigenvalue of $(\mathrm{LCM}_n, \mathrm{GCD}_n)$ if and only if the binary expansion of $n$ begins with "10."

From a computational perspective, detection of $-1$ as a g-eigenvalue can be reduced to testing $\det(L_n + G_n) = 0$ rather than fully expanding the characteristic polynomial.

6. Connections to Classical Interlacing Results and Applications

The classical Cauchy interlacing theorem addresses standard eigenvalues of Hermitian matrices: the eigenvalues of any $k \times k$ principal submatrix interlace those of the full matrix. The present result generalizes this phenomenon to the pencil $(A, B)$, leveraging the generalized Rayleigh quotient

$\frac{x^* A x}{x^* B x}$

and its classical minimax characterization for Hermitian pencils. Generalized eigenvalue interlacing is particularly useful in stability theory, constrained optimization, vibration analysis, and the analysis of arithmetic matrix pencils (e.g., MAX–MIN, LCM–GCD). Knowledge of interlacing facilitates inductive spectral arguments and provides bounds and structure for the spectra of principal subpencils, making it a foundational tool in matrix analysis (Merikoski et al., 23 Jan 2026).