notes on matrix polynomials (minimal polynomial)
A matrix is nonderogatory if every eigenvalue has geometric multiplicity 1.
Theorem: A is nonderogatory iff that commutes with A is a polynomial in A. (Claim: If S^{-1} B S = f(J), then B = f(A).) (The inverse is not proved on textbook.)
Theorem: Every matrix A can be written A=AD+AN, where AD diagonalizable, AN nilpotent, and AD and AN commute.
Corollary: Am→0, as m→∞ iff |λ|<1 for any eigenvalue of A.
An annihilatory polynomial is any g∈C[t] s.t. g(A)≡0.
The minimal polynomial of A, mA(t), is the unique monic (首一) polynomial of minimum positive degree that annihilates A.
Theorem: mA is well defined.
(Proof: polynomial division)
Corollary: The minimal polynomial divides the characteristic polynomial and has all the same roots.
Property: TFAE:
Let monic polynomial f(t)=tn+an−1tn−1+⋯+a0, the companion matrix of f is [0−a0I−an−1].
Property: Every monic polynomial is the minimal and characteristic polynomial of its companion.
The set of polynomials in one variable t over C is a vector space.