MAT 242 Chap.10 Diagonalization, Matrix Powers and Jordan Form
Diagonalization, Matrix Powers and Jordan Form
The diagonalization test
An eigenvalue's multiplicity is its exponent in the factored characteristic polynomial, and its eigenspace dimension can be smaller but never larger. A matrix is diagonalizable exactly when every eigenspace reaches the full multiplicity. Then A = PDP⁻¹, where P holds the eigenspace basis vectors as columns and D is diagonal with each eigenvalue in the position of its eigenvector.
If even one eigenvalue falls short, the matrix is not diagonalizable, and the explanation should name that eigenvalue with both numbers.
Why it is worth doing
Powers become easy. In a product of copies of PDP⁻¹ the inner P⁻¹P pairs cancel, leaving PDᵏP⁻¹, and a diagonal matrix is powered entry by entry.
The paid chapter derives a closed formula for every power of a two-by-two matrix and checks it against direct multiplication, with a warning about keeping parentheses around negative bases so that alternating signs survive.
When the test fails
The course notes include an optional section on generalized eigenvectors.
When an eigenspace is too small, chains of vectors, each mapped by A − λI to the previous one, replace the missing eigenvectors. Gluing the chains into P in order gives A = PTP⁻¹ with T nearly diagonal: eigenvalues on the diagonal and 1s just above it wherever two columns are linked.
The rule the notes stress is simple: never break a chain.
On tests, the decision itself is the core skill: eigenspace dimensions compared with multiplicities, followed by either P and D or the reason they do not exist.
How sample solutions grade this
A diagonalization question on the sample tests has two parts.
The first computes eigenvalues and eigenspaces; the second answers yes, with P and D, or no, with an explanation. The explanation earns its credit only when it names the eigenvalue whose eigenspace is too small and states both numbers.
The sample key also gives full credit to a second part that is consistent with an error made in the first, which is one more reason to show every step.
Checking a factorization
After building P and D, the product PDP⁻¹ should reproduce A.
A cheaper check avoids the inverse altogether: AP should equal PD, because each column of AP is A times an eigenvector and each column of PD is that eigenvector scaled by its eigenvalue. Comparing one column at a time also pinpoints which eigenvector, if any, was computed wrongly. Remember that D must list the eigenvalues in the same order as the columns of P.
What this chapter covers
- 01
Multiplicity versus eigenspace dimension
- 02
Building P and D
- 03
Matrix powers through PDⁿP⁻¹
- 04
Explaining a non-diagonalizable matrix
- 05
Generalized eigenvectors and Jordan blocks
Worked example · free
Decide whether a matrix is diagonalizable
- 1A is triangular, so its only eigenvalue is 3, with multiplicity 2.
- 2A − 3I = [0 1; 0 0] has one pivot and one free variable, so the eigenspace has dimension 1.
- 2Dimension 1 is less than multiplicity 2, so A is not diagonalizable.
Key terms
- Multiplicity
- Multiplicity is the exponent of a factor (λ − r) in the characteristic polynomial.
- Diagonalizable Matrix
- A diagonalizable matrix can be written as PDP⁻¹ with D diagonal and P built from eigenvectors.
- Jordan Block
- A Jordan block is a square matrix with one eigenvalue down its diagonal and 1s directly above it.
Diagonalization, Matrix Powers and Jordan Form FAQ
Does the order of eigenvalues in D matter?
Any order works, provided each eigenvalue sits in the same position as its eigenvector's column in P. Reordering one without the other gives a wrong factorization.
Why is A to the power k equal to P D to the power k P inverse?
Writing the k copies of PDP⁻¹ side by side, every P⁻¹ meets a P and cancels to the identity, leaving P, then k copies of D, then P⁻¹.
Is every matrix with repeated eigenvalues non-diagonalizable?
No. A repeated eigenvalue is fine when its eigenspace reaches the full multiplicity; the identity matrix is the simplest example. Only an eigenspace that falls short blocks diagonalization.
Exam move
Make a two-column table for each matrix, multiplicity against eigenspace dimension, before writing P. The table itself is most of the explanation a grader wants. Practise the verdict on matrices that are diagonalizable and ones that are not in equal numbers, so the decision is made from dimensions rather than habit.
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