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The Jacobi matrix spectrum is the set of Gauss nodes

Statement

Let $(a, b, m, r, μ)$ be moment data from Positive-measure Gaussian quadrature moment data with $r \geq 2 m - 1$ . For $0 \leq j < m$ , let $h_{j}, α_{j}$ be as in Monic orthogonal polynomials satisfy a three-term recurrence, and for $1 \leq j < m$ let $β_{j} = h_{j} / h_{j - 1}$ . Put $β_{0} = 0$ and define the real symmetric tridiagonal Jacobi matrix $J_{m} = (\begin{array}{cccc} α_{0} & \sqrt{β_{1}} \\ \sqrt{β_{1}} & α_{1} & ⋱ \\ ⋱ & ⋱ & \sqrt{β_{m - 1}} \\ \sqrt{β_{m - 1}} & α_{m - 1} \end{array}),$ with the evident one-by-one interpretation for $m = 1$ . If $a < x_{1} < \dots < x_{m} < b$ are the roots of $p_{m}$ , then $spec (J_{m}) = {x_{1}, \dots, x_{m}}$ , where spectrum denotes the set of eigenvalues as in Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism; every eigenvalue is simple. If $q_{j} : = p_{j} / \sqrt{h_{j}}$ for $0 \leq j < m$ , then $(q_{0} (x_{i}), \dots, q_{m - 1} (x_{i}))^{T}$ is a nonzero eigenvector of $J_{m}$ for $x_{i}$ .

Facts & Assumptions

Given: Moment data and the monic orthogonal polynomials through degree $m$ .

[F1]

Monic orthogonal polynomials satisfy a three-term recurrence: $x p_{j} = p_{j + 1} + α_{j} p_{j} + β_{j} p_{j - 1}$ for $1 \leq j < m$ , and $x p_{0} = p_{1} + α_{0} p_{0}$ .

[F2]

The Gauss orthogonal polynomial has simple roots inside the interval: the degree- $m$ polynomial $p_{m}$ has exactly $m$ distinct roots $x_{i} \in (a, b)$ .

[F3]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis.

[F4]

The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products and In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose: a real symmetric matrix is self-adjoint for the standard Euclidean inner product on $R^{m}$ .

[F5]

Nonnegative real square roots exist uniquely: every positive real $β_{j}$ has a unique positive square root.

[F6]

Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism: the spectrum of a finite-dimensional endomorphism is its set of eigenvalues.

Proof

1.1F1F2F5algebra

By [F1] and [F5], the normalized polynomials $q_{j} : = p_{j} / \sqrt{h_{j}}$ satisfy $x q_{j} = \sqrt{β_{j + 1}} q_{j + 1} + α_{j} q_{j} + \sqrt{β_{j}} q_{j - 1}$ for $0 \leq j < m - 1$ , with $q_{- 1} = 0$ . For the final index, the monic recurrence gives $x p_{m - 1} = p_{m} + α_{m - 1} p_{m - 1} + β_{m - 1} p_{m - 2}$ ; at a root $x_{i}$ the $p_{m}$ term vanishes. After dividing by $\sqrt{h_{m - 1}}$ , this is the final row of $J_{m} z_{i} = x_{i} z_{i}$ for $z_{i} = (q_{0} (x_{i}), \dots, q_{m - 1} (x_{i}))^{T}$ . When $m = 1$ , the identity $x p_{0} = p_{1} + α_{0} p_{0}$ gives the same conclusion at the sole root.

2.1F2step 1.1F5algebra

The vector $z_{i}$ is nonzero because $q_{0} = 1 / \sqrt{h_{0}}$ and $h_{0} = μ ([a, b]) > 0$ . Thus each of the $m$ distinct roots in [F2] is an eigenvalue of $J_{m}$ .

3.1F2F3F4F6step 2.1∎

The matrix $J_{m}$ is real symmetric by its definition, so [F4] makes its endomorphism self-adjoint on $R^{m}$ . By [F3], it has an orthonormal eigenbasis of $m$ vectors; hence it has exactly $m$ eigenvalues counted with multiplicity. The $m$ distinct eigenvalues already exhibited in step 2.1 exhaust this list, so each is simple and there are no others. By [F6], this eigenvalue set is $spec (J_{m})$ .

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Sources

John A. Gubner, Gaussian Quadrature and the Eigenvalue Problem, §§2–5, 2020 (standard reference, not scraped)
MATH 373 Lecture Notes, Lecture 14: Gaussian Quadrature, §§10.2–10.5 (standard reference, not scraped)
Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)