Linear Algebra
Chapter 6/Eigenvalues

6.3Diagonalization

The closing idea of the course: if has enough eigenvectors, changing to the eigenvector basis makes diagonal. Powers, exponentials, and differential equations then become scalar problems applied one eigenvalue at a time.

What you must be able to do

  • is diagonalizable iff it has linearly independent eigenvectors. Then with eigenvectors as columns of and eigenvalues on 's diagonal — in matching order.
  • distinct eigenvalues ⟹ diagonalizable (sufficient, not necessary). Repeated eigenvalues require checking geometric multiplicity.
  • Defective = fewer than independent eigenvectors = not diagonalizable.
  • and with .
Theorem 6.3.1
Eigenvectors belonging to distinct eigenvalues are linearly independent.
Definition · Diagonalizable
is diagonalizable if there exist a nonsingular and diagonal with , equivalently . We say diagonalizes .
Theorem 6.3.2 · The characterisation
An matrix is diagonalizable if and only if has linearly independent eigenvectors.

Proof idea: says column-by-column that . is invertible exactly when its columns — the eigenvectors — are independent.

  1. 1The columns of are eigenvectors; the diagonal of holds the matching eigenvalues, in the same order.
  2. 2 is not unique — reorder the columns (reorder to match) or rescale any column.
  3. 3 distinct eigenvalues ⟹ diagonalizable. With repeats, it depends on whether the eigenspaces are big enough.
  4. 4If has fewer than independent eigenvectors it is defective, and no such exists.
!Exam trap
Diagonalizability and nonsingularity are completely independent. is singular and diagonalizable; is nonsingular and defective. Diagonalizability is about the supply of eigenvectors, not about .
Worked example

Diagonalizing a matrix

0/5 steps
Find and with .

Defective matrices

Example 4 of the notes makes the point sharply. Both matrices below have eigenvalues , , yet only one is diagonalizable:

★Key idea
Repeated eigenvalues are the only place defectiveness can hide. For each repeated , compute and compare it with the algebraic multiplicity. Equal for all eigenvalues ⟹ diagonalizable.

Powers and the matrix exponential

Definition · Matrix exponential
For any square , . For a diagonal this is , and for a diagonalizable the series telescopes to
Worked example

Computing

0/5 steps
Compute the matrix exponential.
∞Closing the loop with §6.2
For diagonalizable , with . The matrix-exponential formula and the eigenvector formula from §6.2 are the same answer — and the second is just the first written out in the eigenbasis.
Check your work

Eigenvalues, eigenspaces & diagonalizability

Builds the characteristic polynomial exactly, factors out rational roots, then solves the null space of A − λI.

characteristic polynomial
p(λ) = λ³ − 2λ² + λ = 0
trace = Σλ
2
det = Πλ
0
diagonalizable
yes
λ = 0algebraic mult. 1 · geometric mult. 1
eigenspace basis:
1
1
1
λ = 1algebraic mult. 2 · geometric mult. 2
eigenspace basis:
3
1
0
-1
0
1

Check yourself

Q1
An matrix is diagonalizable if and only if:
T / F
If is diagonalizable then is nonsingular.
Q3
A matrix has eigenvalues and . The eigenspace for turns out to be one-dimensional. What follows?
Q4
If , then equals:
T / F
For a diagonalizable , the solution of , can be written .