5.2

• Review the box
• Algebraic multiplicity bounds geometric multiplicity. Take a basis for the eigenspace. Extend.
• Let T be a linear operator and λ_i be distinct eigenvalues. For each λ_i, let v_i ∈ E_λi, then if ∑v_i = 0, each v_i is zero.
• Unions of linearly independent subsets from eigenspaces are still linearly independent.
• Diagonalizable if and only if algebraic multiplicity equals geometric multiplicity.

  • Proof: Let m_i be algebraic multiplicity, d_i be geometric multiplicity, and n = dim V.

    Suppose diagonalizable. Then let B be a basis consisting of eigenvectors. Intersect this basis with each eigenspace. Call it B_i. Let n_i = B_i. Then n ≤ ∑n_i ≤ ∑d_i ≤ ∑m_i ≤ n.

    So ∑(m_i − d_i) = 0. But each term must be nonnegative.

    Conversely, suppose the multiplicties are equal. Then take union of eigenbasis. They form a basis for full space.

• Testing diagonalizability. Check splitting, check multiplicities. This also computes the diagonalization along the way.
• What are eigenvalues of derivative?
• Finite dimensional matrices always have eigenvalues if you extend the field.
• Give example of infinite dimension transform with no eigenvalues.
• Work through example 6 in book.

Direct sums

• Define direct sum
• Give example
• Show Jasper’s example to motivate
• Diagonalizable if and only if direct sum of eigenspaces
• The following are equivalent:
  • direct sum
  • sums and unique way of writing 0
  • unique way of writing any element
  • union of ordered basis for summands is ordered basis
  • exists order basis fof summands so that union is ordered basis.
• Do more examples