Loewner Matrix: Theory and Applications

• Loewner matrix is a structured matrix constructed from divided differences that encodes properties like operator monotonicity and convexity.
• It reveals essential spectral and inertia characteristics, providing insights into positive definiteness and sensitivity to noise.
• The framework underpins efficient data-driven model reduction and system realization for both finite and infinite-dimensional systems.

A Loewner matrix is a structured matrix constructed from the divided differences of a real function, typically on positive numbers, and plays a fundamental role in matrix analysis, operator theory, and modern data-driven model reduction. The Loewner matrix encodes key properties of the underlying function—such as operator monotonicity and convexity—and underpins data-driven realization frameworks for high-dimensional and infinite-dimensional systems.

1. Definition and Classical Properties

Given distinct positive real numbers $p_1 < p_2 < \cdots < p_n$ and a real parameter $r$, the prototypical Loewner matrix $L_r$ is the $n \times n$ real matrix with entries

$$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$

This matrix is the divided-difference matrix of $f(t) = t^r$ at the nodes $p_i$, and the extension to the diagonal via the limit yields continuity and Hermitian structure.

Loewner’s classical theorem asserts that for $0 < r < 1$, $L_r$ is positive definite for all choices of $p_i$. More generally, for any continuously differentiable $r$0, the Loewner matrix $r$1 is defined via first divided differences, $r$2, capturing the local linear behavior of $r$3 across the spectral data.

2. Inertia of Loewner Matrices

The inertia of a Hermitian matrix $r$4 is the triple $r$5: numbers of positive, zero, and negative eigenvalues. The complete inertia of $r$6 for all real $r$7 is given by the following results [{(Bhatia et al., 2015)}]:

• $r$8 is singular if and only if $r$9. For these $L_r$0, the size and position of the zero block in the inertia is explicit.
• For non-integer $L_r$1 with $L_r$2, writing $L_r$3 or $L_r$4:
  • If even: $L_r$5,
  • If odd: $L_r$6,
• For $L_r$7, $L_r$8 does not change with $L_r$9.
• Every nonzero eigenvalue of $n \times n$0 is simple.

For $n \times n$1, $n \times n$2 has inertia $n \times n$3: exactly one positive eigenvalue and $n \times n$4 negative ones. The sign pattern flips across integer values of $n \times n$5.

3. Operator Monotonicity, Convexity, and Conditional Definiteness

Loewner matrices encode operator-theoretic properties of functions. Charles Loewner’s theorem states $n \times n$6 is operator-monotone on $n \times n$7 if and only if all $n \times n$8 are positive semidefinite, for all $n \times n$9 and points $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$0 [{(Hiai et al., 2010)}].

Matrix convexity and monotonicity can be sharply characterized in terms of “conditionally positive definite” (c.p.d.) or “conditionally negative definite” (c.n.d.) Loewner matrices:

• $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$1 c.n.d. on the subspace $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$2 implies matrix convexity.
• $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$3 c.p.d. on the same implies matrix monotonicity.

These results are particularly transparent for $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$4:

• $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$5 is operator monotone on $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$6 iff $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$7.
• $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$8 is c.p.d. for all $$(L_r)_{ij} = \begin{cases} \displaystyle\frac{p_i^r - p_j^r}{p_i - p_j}, & \text{if } i \neq j \ r p_i^{r-1}, & \text{if } i = j \end{cases}$$9 iff $f(t) = t^r$0 or $f(t) = t^r$1; c.n.d. iff $f(t) = t^r$2 or $f(t) = t^r$3 [{(Hiai et al., 2010)}].

4. Loewner Framework in Data-driven Model Realization

Modern systems theory leverages Loewner matrices to perform model reduction and system identification from input-output (frequency response) data. Given left and right sets of frequency points $f(t) = t^r$4, tangential directions, and associated data, the Loewner $f(t) = t^r$5 and shifted Loewner $f(t) = t^r$6 matrices are defined by Cauchy-type formulas [{(Palitta et al., 2021)}, {(Gosea et al., 2017)}]: $f(t) = t^r$7

$f(t) = t^r$8

The Loewner pencil $f(t) = t^r$9 provides the data-driven counterpart of the transfer function realization, and its singular value decomposition determines the minimal reduced-order model. The Cauchy-like structure enables highly efficient $p_i$0 algorithms via Hierarchically Semi-Separable (HSS) representations, circumventing $p_i$1 storage bottlenecks for massive datasets [{(Palitta et al., 2021)}].

In the multivariate (parameterized) setting, the n-dimensional Loewner matrix is constructed from divisors over all $p_i$2 variables. Its null space is computable by recursive application of 1-D Loewner problems, yielding for $p_i$3 a complexity $p_i$4 versus $p_i$5 for direct linear algebra, and facilitating barycentric rational approximants in high dimension [{(Antoulas et al., 2024)}].

5. Spectral Properties, Sensitivity, and Robustness

Loewner matrices’ spectral profile determines robustness of realized system poles to noise and data selection. Fast singular value decay of the Loewner matrix (from large separation of left/right interpolation points) signals potential ill-conditioning and high sensitivity of realized poles. Interleaving data near the system’s true spectrum typically yields better conditioned and more robust pencils [{(Embree et al., 2019)}].

Explicit factorization of the Loewner pencil in terms of control-theoretic Krylov matrices produces a minimal system realization: if $p_i$6 is the Loewner pencil, then $p_i$7, $p_i$8, $p_i$9, and $0 < r < 1$0 are constructed directly from data. Analyses yield two classes of pole sensitivity:

• Unstructured (worst case) condition number depends on the norm and conditioning of the left/right Krylov projections.
• Structured sensitivity (to measurement noise) further depends on the Cauchy structures and actual noise profile in data [{(Zhang et al., 2021)}].

Pseudospectral analysis quantifies the degree to which noisy measurements or adverse partitioning will displace the recovered system poles. Specific algorithms exploit the Loewner structure to compute pseudospectra efficiently, providing practical diagnostics for model reliability [{(Embree et al., 2019)}].

6. Schur Multiplier Norms and Commutator Inequalities

Loewner matrices also arise in operator norm inequalities for matrix commutators. The linear map $0 < r < 1$1 is a Schur multiplier, and for any Hermitian $0 < r < 1$2 and function $0 < r < 1$3, $0 < r < 1$4 can be written as the Schur product of $0 < r < 1$5 with $0 < r < 1$6. The operator norm of this Schur multiplier provides tight upper bounds for

$0 < r < 1$7

across Schatten $0 < r < 1$8-norms. For operator monotone $0 < r < 1$9, the norm is bounded by $L_r$0. For concave/convex $L_r$1, it can be expressed explicitly using the golden ratio and extremal derivatives $L_r$2 [{(Audenaert, 2013)}].

Such inequalities allow precise control of commutator size and reveal structural properties of matrix functions through their Loewner matrices.

7. Broader Applications and Recent Developments

Loewner matrices play a central role well beyond operator theory:

• In matrix concentration inequalities, the ordering induced by Loewner matrices—via Loewner order and anti-order—yields sharp distinct upper tail bounds for minimum and maximum eigenvalues of random matrices, with new results removing a factor $L_r$3 gap in prior bounds [{(Malekian et al., 2024)}].
• In majorization, matrix means, perturbation theory, and rational function interpolation, the Loewner matrix codifies the divided difference kernel, underlying interpolation formulas and barycentric representations.
• In data-driven system realization and model reduction for infinite-dimensional systems (e.g., damped beams), the Loewner framework outperforms modal truncation and remains robust without explicit finite element discretization, leveraging only input-output data [{(Gosea et al., 2017)}].

The Loewner matrix thus acts as a unifying object in matrix spectral analysis, interpolation, learning, and optimization, continuously informing both the theory and the high-performance computation of system models and matrix functions.