Jodan Canonical Form

Contents

  • Chapter 1: Jordan Canonical Form

  • Differential Equations ITC

Notations

Eigenvalues and Eigenvectors

Definitions

  1. Eigenvalue: A scalar λ ∈ K is an eigenvalue of A ∈ Mn(K) if there exists a nonzero vector x ∈ Kn such that Ax = λx.

  2. Eigenvector: The vector x is the eigenvector associated with the eigenvalue λ.

  3. Spectrum of A: All eigenvalues of A are collectively called the spectrum of A, denoted as spect(A).

  4. Eigenspace: Defined as Eλ = Ker(A − λIn) = {x ∈ Kn : (A − λIn)x = 0}.

  5. A-invariant Subspace: Subspace S of Kn is an A-invariant subspace if {Ax : x ∈ S} ⊆ S.

Theorems

Theorem 1: If λ ∈ K is an eigenvalue of A, then the eigenspace Eλ is a subspace of Kn and A-invariant. The dimension of this eigenspace is called the geometric multiplicity of λ, denoted as gm(λ).

Theorem 2: For A ∈ Mn(K) and λ ∈ K, the following are equivalent:

  1. λ is an eigenvalue of A.

  2. (A − λI)x = 0 has a nontrivial solution.

  3. Eλ ≠ {0}.

  4. det(A − λI) = 0.

Theorem 3: If V is a vector space and λ ∈ K is an eigenvalue of linear operator L, then the eigenspace Eλ is a subspace of V and L-invariant. If V is finite-dimensional, Eλ is finite-dimensional.

Characteristics of Polynomials

  1. Characteristic Polynomial: pL(λ) = det(L − λI) does not depend on a basis for V. The characteristic equation pA(λ) = 0 is crucial for identifying eigenvalues.

  2. Minimal Polynomial: mA(λ) is the monic polynomial of least degree such that mA(A) = 0. mA divides every polynomial having A as a zero.

Diagonalization and Triangularization

Definitions

  1. Diagonalizable: A matrix A ∈ Mn(K) is diagonalizable if A ∼ D, with D being a diagonal matrix.

  2. Triangularizable: A is triangularizable if A ∼ T, with T as a triangular matrix.

Theorems

  • Theorem 17: An operator L is diagonalizable over K if and only if the corresponding matrix A is diagonalizable.

  • Theorem 18: The following assertions relate to diagonalizability of L:

    1. There exists a basis of V formed by eigenvectors of L.

    2. V = Eλ1 ⊕ · · · ⊕ Eλk.

    3. n = dim Eλ1 + ... + dim Eλk.

Cayley-Hamilton Theorem

Definitions

  1. Cyclic subspace generated by x, L(x), ..., is an L-cyclic subspace.

  2. Cayley-Hamilton: pL(L) = 0 for the characteristic polynomial of L.

Theorems

  • Theorem 26: The characteristic polynomial of an L-invariant space divides the characteristic polynomial of L.

  • Theorem 28: If pL(λ) is the polynomial characteristic of L, then pL(L) = 0.

Jordan Canonical Form

Definitions

  1. A linear operator is nilpotent if Lk = 0 for some positive integer k.

  2. For eigenvalue λ:

    • Index of λ: smallest k such that rank(L − λI)k = rank(L − λI)k+1.

    • Generalized eigenvector: A nonzero vector x such that (L − λI)p (x) = 0 for some p.

Theorems

  • Theorem 30: If the characteristic polynomial of L splits over K, then:

    1. dim(Gλ(L)) ≤ k.

    2. Gλ(L) = N((L − λI)k).

  • Theorem 34: If the characteristic polynomial splits over K, there exists an ordered basis for Gλ consisting of disjoint cycles of generalized eigenvectors, proving L is Jordanizable.

Jordan Blocks

  1. Jordan Block: Jb(λ) consists of λ on the diagonal and 1's immediately above the diagonal.

  2. Jordan Segment: Js(λ) is a block matrix with multiple Jordan blocks corresponding to λ.

  3. Jordan Canonical Form: A matrix represented in Jordan form as described above.

Unique Properties

  1. Number of Jordan segments and their size is uniquely determined by the entries of A.

  2. Real Jordan forms involve Jordan blocks corresponding to real numbers and complex conjugate pairs.

This represents a comprehensive overview of the Jordan Canonical Form and associated principles for studying eigenvalues, eigenvectors, and polynomial characteristics.