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A singular leading coefficient gives an infinite polynomial eigenvalue

Example

Take the grade-two matrix polynomial $P (λ) = diag (1, λ^{2})$ , so $n = m = 2$ , $\det P (λ) = λ^{2}$ , and $A_{2} = diag (0, 1)$ is singular. The finite root $λ = 0$ and the infinite root are different projective eigenvalues. Their algebraic and geometric multiplicities are computed directly below.

Facts & Assumptions

Given: $A_{0} = diag (1, 0)$ , $A_{1} = 0$ , and $A_{2} = diag (0, 1)$ for a regular grade-two polynomial.

[F1]

Regular matrix polynomials and homogeneous eigenpairs: $P_{h} (1, 0) = A_{2}$ , and the algebraic multiplicity at infinity is the zero order at $0$ of $\det P_{2}^{r e v}$ .

[F2]

First Frobenius companion pencil of a matrix polynomial: at infinity the homogeneous companion pencil is $- B_{c} = diag (A_{2}, I_{2})$ , with block eigenvectors $(x, 0)^{T}$ for $x \in \ker A_{2}$ .

[F3]

The first Frobenius companion pencil preserves the finite and infinite polynomial spectrum: the finite and infinite eigenspaces correspond to those of the polynomial, and the infinite algebraic multiplicity is $m n - \deg \det P$ .

Verification

technique · direct calculation

1.1givenF1F3algebra

The determinant is $\det P (λ) = λ^{2} \equiv̸ 0$ , so $P$ is regular; its degree is $2$ , while $m n = 4$ . Thus the theorem's degree-defect formula gives infinite algebraic multiplicity $4 - 2 = 2$ . Also $P_{2}^{r e v} (μ) = diag (μ^{2}, 1)$ , whose determinant has a zero of order two at $μ = 0$ .

2.1givenF1F2step 1.1algebra

The leading matrix $A_{2}$ has rank one and kernel $span {e_{1}}$ , so the infinite eigenspace has geometric multiplicity one and is represented in the companion pencil by $(e_{1}, 0)^{T}$ . This differs from its algebraic multiplicity two.

3.1givenF1F2F3step 1.1algebra∎

At the finite root $λ = 0$ , $P (0) = diag (1, 0)$ has kernel $span {e_{2}}$ . The finite lift is $(0, e_{2})^{T}$ , which the companion theorem maps to a pencil eigenvector. Thus the finite root zero is not the infinite eigenvalue represented by $(e_{1}, 0)^{T}$ .

Depends on

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Sources

D. Steven Mackey, Niloufer Mackey, and Françoise Tisseur, Polynomial Eigenvalue Problems: Theory, Computation, and Structure, §2 (standard reference, not scraped)
Françoise Tisseur and Karl Meerbergen, The Quadratic Eigenvalue Problem, §3.1 (standard reference, not scraped)