CAUSE
One question. One truth.
“If W is T-invariant subspace, then why characteristic polynomial of T restricted to W divides characteristic polynomial of T?”
The characteristic polynomial of T restricted to a T-invariant subspace W divides the characteristic polynomial of T because T can be represented in a block upper-triangular form relative to a basis adapted to W, making the characteristic polynomial of T the product of the characteristic polynomials of T|W and its complement.
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CURRENT STATE
Given a linear operator T on a finite-dimensional vector space V and a T-invariant subspace W, the characteristic polynomial of T factors as the product of the characteristic polynomials of T restricted to W and to a complementary T-invariant subspace. This follows from the decomposition V = W ⊕ W' where both are T-invariant, ensuring pT|W divides pT. The Cayley-Hamilton theorem and minimal polynomial divisibility support this factorization.
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