Diagonalization

Choosing the basis in which the matrix is simple.

12 problems · about 24 minutes · N-VM.C.11

What this lesson teaches

The student explains diagonalization and uses it to compute matrix powers.

In the basis of its own eigenvectors, a matrix becomes diagonal: A = PDP⁻¹.
Powers become easy, because a diagonal matrix is raised entry by entry.
A matrix without enough independent eigenvectors, such as a shear, cannot be diagonalised.

Straightforward practice. Get the method working first.

5 problems

An eigenvalue of 3. What is its diagonal entry in D to the power 3?
Answer 27
Why 27.
Why is diagonalization useful?
Answer Repeated application becomes raising each eigenvalue to a power.
Why It makes powers cheap.
An eigenvalue of 2. What is its diagonal entry in D to the power 5?
Answer 32
Why 2 to the fifth.
An eigenvalue of 1. What is its diagonal entry in D to the power 100?
Answer 1
Why 1 to any power.
How many independent eigenvectors does a 2 by 2 need to be diagonalisable?
Answer 2
Why One per dimension.

The same ideas with more to keep track of.

3 problems

In A = PDP inverse, what sits on the diagonal of D? Enter 1 for eigenvalues, 2 for eigenvectors.
Answer 1
Why The vectors go into P.
An eigenvalue of 0.5. What is its diagonal entry in D to the power 2, as a decimal?
Answer 0.25
Why Half squared.
A matrix already diagonal. Is it diagonalisable? 1 yes, 0 no.
Answer 1
Why It is already in that form.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order what happens when a diagonalised matrix acts on a vector.
Answer 1. Translate the vector into eigenvector coordinates 2. Stretch each coordinate by its eigenvalue 3. Translate back to the original coordinates 4. Read off the result
Why The stretching is the easy middle step.
An eigenvalue of −1. What is its diagonal entry in D to the power 4?
Answer 1
Why An even power.
The Power: An eigenvalue of 3. What is the matching diagonal entry of D to the power 4?
Answer 81
Why 81.
The Shear: A shear has only one independent eigenvector. Can it be diagonalised? 1 yes, 0 no.
Answer 0
Why No.