In the basis of its own eigenvectors, a matrix becomes diagonal. All the off-diagonal complication was an artefact of the basis you happened to be using.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The factorisation
A = PDP⁻¹, where the columns of P are the eigenvectors and D holds the eigenvalues on its diagonal.
How to read it
Translate into eigenvector coordinates, stretch each axis by its eigenvalue, translate back. Three simple steps replace one complicated one.
Powers become easy
A to the n equals P Dⁿ P⁻¹, and raising a diagonal matrix to a power just raises each diagonal entry. Fifty multiplications become one.
When it fails
Diagonalization needs enough independent eigenvectors. A shear has only one direction it preserves, so it cannot be diagonalised.
Why anyone cares
Repeated application is everywhere: population models, page ranking, quantum states. Diagonalization is what makes the long run computable.
Choosing the basis that makes it simple
In a basis of eigenvectors, the matrix becomes diagonal — the transformation is just independent scalings along those directions. Diagonalisation is a change of viewpoint, not of the transformation.
The decomposition
A = PDP⁻¹, where P holds the eigenvectors as columns and D the eigenvalues on the diagonal. Reading it right to left, you change basis, scale, and change back.
Powers become trivial
Aⁿ = PDⁿP⁻¹, and raising a diagonal matrix to a power just raises its entries. That converts an expensive computation into an easy one, which matters for iterated systems.
Not every matrix diagonalises
A matrix needs enough independent eigenvectors to form a basis. When it has too few, the Jordan form is the substitute — less clean, and still enough to compute with.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(2, 1)
- j-hat lands on(1, 2)
- Determinant3
- Eigenvalues3 and 1
The dashed lines are the eigen-directions. A vector on one of those lines still points the same way after the transformation, only longer or shorter, and the eigenvalue is that stretch factor.
The shaded parallelogram is the image of the unit square, and its area is 3. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Nadia Take a Tenth Power
Nadia must compute A to the power 10 where A has eigenvalues 2 and 1.
- Step 1
She confirms A has two independent eigenvectors, so it is diagonalisable.
Step 4: Your Turn
Practice makes it stick.
The Power
Problem 1 of 2
An eigenvalue of 3. What is the matching diagonal entry of D to the power 4?
The Shear
Problem 2 of 2
A shear has only one independent eigenvector. Can it be diagonalised? 1 yes, 0 no.
Diagonalize It
1 of 8
An eigenvalue of 2. What is its diagonal entry in D to the power 5?
2 of 8
An eigenvalue of 1. What is its diagonal entry in D to the power 100?
3 of 8
How many independent eigenvectors does a 2 by 2 need to be diagonalisable?
4 of 8
In A = PDP inverse, what sits on the diagonal of D? Enter 1 for eigenvalues, 2 for eigenvectors.
5 of 8
An eigenvalue of 0.5. What is its diagonal entry in D to the power 2, as a decimal?
6 of 8
A matrix already diagonal. Is it diagonalisable? 1 yes, 0 no.
7 of 8
Order what happens when a diagonalised matrix acts on a vector.
- 1Stretch each coordinate by its eigenvalue
- 2Translate back to the original coordinates
- 3Read off the result
- 4Translate the vector into eigenvector coordinates
8 of 8
An eigenvalue of −1. What is its diagonal entry in D to the power 4?
Step 5: Quick Check
Show what you know.
Question 1 of 2
An eigenvalue of 3. What is its diagonal entry in D to the power 3?
Question 2 of 2
Why is diagonalization useful?
What You Learned
- 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.