# 5.2

• Review the box
• Algebraic multiplicity bounds geometric multiplicity. Take a basis for the eigenspace. Extend.
• Let T be a linear operator and λi be distinct eigenvalues. For each λi, let vi ∈ Eλi, then if vi = 0, each vi 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 mi be algebraic multiplicity, di 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 Bi. Let ni = Bi. Then n ≤ ∑ni ≤ ∑di ≤ ∑mi ≤ n.

So ∑(mi − di) = 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