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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: Am0, as m iff |λ|<1 for any eigenvalue of A.

An annihilatory polynomial is any gC[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:

  1. A diagonalizable
  2. m_A splits into distinct linear factors
  3. every root of m_A has multiplicity 1
  4. any t_0 s.t. m_A(t_0) =0, m^1_A(t_0) != 0

Let monic polynomial f(t)=tn+an1tn1++a0, the companion matrix of f is [0a0Ian1].

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.


🏷 Category=Algebra Category=Matrix Analysis