Diagonalization

Choosing the basis in which the matrix is simple.

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

NameDate

Figure — use these to answer the problems

Two dashed lines means two independent eigen-directions, so this matrix can be diagonalised. Set it to a shear, 1, 1, 0, 1, and only one line survives.

i-hat lands on(2, 1)
j-hat lands on(1, 2)
Determinant3
Eigenvalues3 and 1

Figure 1

Warm Up

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
2
Why is diagonalization useful?
- Repeated application becomes raising each eigenvalue to a power.
- It makes the matrix smaller.
An eigenvalue of 2. What is its diagonal entry in D to the power 5?
Answer
An eigenvalue of 1. What is its diagonal entry in D to the power 100?
Answer
How many independent eigenvectors does a 2 by 2 need to be diagonalisable?
Answer

Build It Up

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
An eigenvalue of 0.5. What is its diagonal entry in D to the power 2, as a decimal?
Answer
A matrix already diagonal. Is it diagonalisable? 1 yes, 0 no.
Answer

Stretch Yourself

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

4 problems

9
Order what happens when a diagonalised matrix acts on a vector.
Write 1 to 4 in the boxes to put these in order.
- Translate the vector into eigenvector coordinates
- Stretch each coordinate by its eigenvalue
- Translate back to the original coordinates
- Read off the result
An eigenvalue of −1. What is its diagonal entry in D to the power 4?
Answer
The Power
An eigenvalue of 3. What is the matching diagonal entry of D to the power 4?
Answer
The Shear
A shear has only one independent eigenvector. Can it be diagonalised? 1 yes, 0 no.
Answer