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The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks
Statement
Let and . For either block-triangular matrix
one has , including or .
Facts & Assumptions
Given: Square diagonal blocks and a compatible off-diagonal block over .
Characteristic polynomials are determinants of minus the matrix, with the empty-block value (For , the characteristic polynomial is when , with for the unique matrix).
The determinant is the signed sum over permutations of products selecting one entry in each column and row (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix).
The polynomial ring is a commutative ring, so [L2] applies to matrices over (Polynomial convolution makes a commutative ring containing as its constant subring).
Proof
Suppose first that is block upper triangular. In a nonzero Leibniz term for , a permutation cannot send an index from the second block into the first: if it did, bijectivity would force some first-block index into the second, where the lower-left block is zero. Thus every surviving permutation preserves both index blocks.
The surviving permutation splits uniquely into one permutation of each block, and its sign and product split accordingly. The determinant sum therefore factors as .
The lower-triangular case is identical with the two block directions interchanged. If either block has size zero, its characteristic polynomial is and the identity reduces to the other block.
Hence both block-triangular forms have the claimed characteristic polynomial.
Depends on
- For $A\in M_n(F)$, the characteristic polynomial is $\chi_A(x)=\det(xI_n-A)$ when $n\geq1$, with $\chi_A(x)=1$ for the unique $0\times0$ matrix
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
- Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring
Used by
- The product of the invariant factors is the characteristic polynomial Corollary
- The identity and a nontrivial Jordan block have the same characteristic polynomial but different minimal polynomials Example
- FALSE: The characteristic polynomial determines whether an endomorphism is diagonalisable False statement
- The complex orientation of the underlying real bundle Lemma
- For invariant W, χ_T=χ_T|_Wχ_T̄ Proposition
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then χ_p(T)(y)=∏_i<n(y-p(λᵢ)) for every p∈ F[x]: the eigenvalues of p(T) are p(λᵢ), counted with algebraic multiplicity Theorem
- Jordan form over the base field exists exactly when the characteristic polynomial splits Theorem
- The geometric multiplicity of an eigenvalue does not exceed its algebraic multiplicity Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- H. Pinkham, Linear Algebra, §12.1.3 (standard reference, not scraped)