DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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First Frobenius companion pencil of a matrix polynomial

Definition

Let $P (λ) = \sum_{j = 0}^{m} A_{j} λ^{j}$ be as in Regular matrix polynomials and homogeneous eigenpairs. For $m \geq 1$ , define $m n \times m n$ block matrices $A_{c}, B_{c}$ by $A_{c} : = [\begin{array}{cccc} A_{m - 1} & A_{m - 2} & \dots & A_{0} \\ - I_{n} & 0 & \dots & 0 \\ 0 & - I_{n} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & - I_{n} & 0 \end{array}], B_{c} : = [\begin{array}{cccc} - A_{m} & 0 & \dots & 0 \\ 0 & - I_{n} & \dots & 0 \\ ⋮ & ⋱ & ⋱ & ⋮ \\ 0 & \dots & 0 & - I_{n} \end{array}] .$ The first Frobenius companion pencil with the sign convention used here is $L (λ) : = A_{c} - λ B_{c} = [\begin{array}{cccc} λ A_{m} + A_{m - 1} & A_{m - 2} & \dots & A_{0} \\ - I_{n} & λ I_{n} & \dots & 0 \\ 0 & - I_{n} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & λ I_{n} \end{array}] .$ For a vector $x \in C^{n}$ and finite $λ$ , its companion lift is $z (λ, x) : = (λ^{m - 1} x, λ^{m - 2} x, \dots, λ x, x)^{T} \in C^{m n}$ ; when $m = 1$ , this is simply $x$ .

Depends on

Regular matrix polynomials and homogeneous eigenpairs

Used by

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Sources

D. Steven Mackey, Niloufer Mackey, and Françoise Tisseur, Polynomial Eigenvalue Problems: Theory, Computation, and Structure, §3 (standard reference, not scraped)
Françoise Tisseur and Karl Meerbergen, The Quadratic Eigenvalue Problem, §3.4 (standard reference, not scraped)