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The homogeneous determinant factors across a block upper triangular pencil

Statement

Let $n_{1}, n_{2} \geq 1$ , $n = n_{1} + n_{2}$ , and let $A, B \in C^{n \times n}$ be block upper triangular with the same two-block partition of sizes $n_{1}, n_{2}$ , with diagonal blocks $A_{11}, A_{22}$ and $B_{11}, B_{22}$ . For every $α, β \in C$ , $\det (β A - α B) = \det (β A_{11} - α B_{11}) \det (β A_{22} - α B_{22}) .$

Facts & Assumptions

Given: A block upper triangular pair with a common two-block partition.

[F1]

For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix: the determinant is the signed finite sum over permutations.

Proof

1.1givenF1algebra

The matrix $M = β A - α B$ is block upper triangular, so every entry in its lower-left block is zero. In a nonzero Leibniz term, each of the first block's columns must therefore be paired with one of the first block's rows. Since the block sizes agree, these choices are a permutation of exactly those rows, and the remaining columns are paired with the remaining rows. Thus every nonzero permutation term splits into one term from each diagonal block.

2.1step 1.1F1algebra∎

Summing the split terms in [F1] gives the product of the two signed permutation sums for $M_{11}$ and $M_{22}$ . These sums are their determinants, which proves the identity.

Depends on

For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix

Used by

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Sources

Daniel Kressner, Numerical Methods and Software for General and Structured Eigenvalue Problems (PhD dissertation, 2004), Chapter 2: The QZ Algorithm (standard reference, not scraped)
Peter Arbenz, Lecture Notes on Solving Large Scale Eigenvalue Problems, §12.8 (standard reference, not scraped)