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Coprime Factors, CRT, and Spectral Projections

When the minimal polynomial factors into coprime pieces, the space splits as a direct sum of T-invariant blocks. We build the spectral projections by Lagrange interpolation — the algebraic heart of the functional calculus.

Coprime factorization splits the space

Suppose m_T = f g with f and g coprime (gcd(f, g) = 1). By Bezout there are polynomials a, b with a f + b g = 1. Evaluating at T: a(T) f(T) + b(T) g(T) = I. Define P = b(T) g(T) and Q = a(T) f(T); then P + Q = I. This is the operator form of coprime factorization and CRT.

P and Q are complementary projections: P^2 = P, Q^2 = Q, P Q = 0. Their images are V_f = ker f(T) and V_g = ker g(T), and V = V_f (+) V_g. Each summand is T-invariant, because P and Q are polynomials in T and polynomials in an operator commute with it. We have factored the operator's domain into independent invariant blocks.

Many factors: spectral projections by interpolation

When m_T(x) = prod (x - lambda_i) splits into DISTINCT linear factors (the diagonalizable case, guide 5), the pieces (x - lambda_i) are pairwise coprime, so V = (+)_i ker(T - lambda_i I): a direct sum of eigenspaces. We can write each projection P_i onto the lambda_i-eigenspace as a single polynomial in T using Lagrange interpolation.

Distinct eigenvalues lambda_1, ..., lambda_r.

Lagrange basis polynomial for lambda_i:
  L_i(x) = prod_{j != i} (x - lambda_j) / (lambda_i - lambda_j)

It satisfies L_i(lambda_j) = 1 if j == i, else 0.

Spectral projection onto the lambda_i eigenspace:
  P_i = L_i(T)

Properties:  P_i P_j = 0 (i != j),  sum_i P_i = I,
             T = sum_i lambda_i P_i   (spectral resolution)
Lagrange interpolation manufactures the spectral projections directly as polynomials in T.

The identity T = sum_i lambda_i P_i is the spectral resolution. It mirrors the spectral theorem of Vol I but needs NO inner product — only that m_T splits into distinct linear factors. Geometry (orthogonal projections) is replaced by pure algebra (interpolation projections).

Functional calculus: defining f(T) sensibly

With distinct eigenvalues, any function f defined on the spectrum gives an operator f(T) = sum_i f(lambda_i) P_i. This polynomial functional calculus turns scalar functions into operators consistently: it agrees with ordinary polynomial substitution, and (fg)(T) = f(T) g(T). You can build sqrt(T), exp(T), or T^{-1} just by acting on eigenvalues.