Module IV·Article I·~1 min read
Invariant Subspaces and the Jordan Form
Linear Operators and the Jordan Form
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The Problem of Diagonalization
Not every operator is diagonalizable. The matrix [[1,1],[0,1]] has a unique eigenvalue 1 of multiplicity 2, but only one eigenline—the space of eigenvectors is one-dimensional. Diagonalization is impossible.
What can be done in this case? Reduce to an almost diagonal form—the Jordan canonical form.
Jordan Block and Jordan Form
A Jordan block J(λ, k) is a k×k matrix of the following type: λ on the diagonal, 1 above the diagonal, 0 everywhere else.
J(λ, 1) = (λ) — a scalar. J(λ, 2) = [[λ,1],[0,λ]]. J(λ, 3) = [[λ,1,0],[0,λ,1],[0,0,λ]].
Theorem on the Jordan Canonical Form: For any linear operator A over an algebraically closed field (for example, ℂ) there exists a basis in which A has a block diagonal form: J(λ₁,k₁) ⊕ J(λ₂,k₂) ⊕ ... This form is unique up to the order of the blocks.
Nilpotent Operators
An operator N is nilpotent if Nᵏ = 0 for some k. A nilpotent operator has a Jordan form with λ = 0.
Examples: the matrix [[0,1,0],[0,0,1],[0,0,0]] is nilpotent, N³ = 0.
A general operator A = D + N (Dunford decomposition): D is diagonalizable (the semisimple part), N is nilpotent, DN = ND.
Minimal Polynomial
The minimal polynomial μₐ(t) is the monic polynomial of least degree such that μₐ(A) = 0.
μₐ(t) divides the characteristic polynomial pₐ(t). The multiplicity of the root λ in μₐ ≤ its multiplicity in pₐ.
A is diagonalizable ⟺ μₐ has no multiple roots.
Computing Functions of Matrices
The Jordan form allows one to compute f(A) for analytic f. For example, eᴬ = P·eᴶ·P⁻¹, where eᴶ is block diagonal with blocks e^(J(λ,k)).
e^(J(λ,k)) = e^λ · [[1,1,1/2!,...,1/(k-1)!],[0,1,1,...],[...],[0,...,0,1]].
Application: solution of systems of ODEs x' = Ax with the matrix exponential x(t) = e^(A t) x(0).
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