Triangularization

TheoremTriangularization of an endomorphism with split characteristic polynomial#✓fully proved

Let \(V\) be a finite-dimensional vector space over a field \(K\), let \(f\) be an endomorphism of \(V\) whose characteristic polynomial splits over \(K\), and let \(\iota\) be a finite linearly ordered set with exactly \(\dim_K V\) elements. Then there is a basis \((b_i)_{i \in \iota}\) of \(V\) in which the matrix of \(f\) is upper triangular: its \((i, j)\) entry is \(0\) whenever \(j < i\).

Proof: induction on dimension. An eigenvector spans an invariant line, the induced map on the quotient has split minimal polynomial (it divides that of \(f\)), and a triangularizing basis of the quotient lifts, together with the eigenvector, to one of \(V\). This is the eigenvector-deflation induction of the proof of Schur's theorem in R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press (2013), Theorem 2.3.1, without unitarity.

theorem exists_basis_upperTriangular_toMatrix
    {ι : Type*} [Fintype ι] [DecidableEq ι] [LinearOrder ι]
    (f : Module.End K V) (h : f.charpoly.Splits)
    (hdim : Module.finrank K V = Fintype.card ι) :
    ∃ b : Module.Basis ι K V, (LinearMap.toMatrix b b f).IsUpperTriangular

Sources

• S. Axler, Linear Algebra Done Right, 4th ed., Springer (2024), Theorem 5.41.

Used by 1

Used byTriangularization of a matrix with split characteristic polynomial

TheoremTriangularization of a matrix with split characteristic polynomial#✓fully proved

Let \(K\) be a field, let \(n\) be a finite linearly ordered index set, and let \(A\) be a square matrix over \(K\) with rows and columns indexed by \(n\) whose characteristic polynomial splits over \(K\). Then there are an invertible matrix \(P\) and a matrix \(T\), both over \(K\) and indexed by \(n\), such that \(T\) is upper triangular (\(T_{ij} = 0\) whenever \(j < i\)) and \(A = P T P^{-1}\).

Proof: triangularize the endomorphism of \(K^n\) given by \(A\) in the standard basis, and conjugate by the change-of-basis matrix from the standard basis to the triangularizing basis.

theorem exists_upperTriangular_similar_of_splits
    {n : Type*} [Fintype n] [DecidableEq n] [LinearOrder n]
    (A : Matrix n n K) (h : A.charpoly.Splits) :
    ∃ P T : Matrix n n K, IsUnit P ∧ T.IsUpperTriangular ∧ A = P * T * P⁻¹

Sources

• R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press (2013), Theorem 2.3.1.

Uses 1

Proof usesTriangularization of an endomorphism with split characteristic polynomial

2 fully proved

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