Diagonalization
Write A = PDP⁻¹ with D diagonal.
Key idea
Write A = PDP⁻¹ with D diagonal.
In depth
A = PDP⁻¹ where D is diagonal with eigenvalues and P has eigenvectors as columns. Possible when there are n independent eigenvectors. It decouples the transformation into independent scalings.
Worked example
A = [4 1; 2 3] has eigenpairs (5, (1, 1)) and (2, (1, −2)). A = PDP⁻¹ with D = diag(5, 2).
Formula
Real-world uses
- Simplifying repeated calculations, such as applying a matrix many times in population models
Common mistake
Assuming all matrices are diagonalizable; some are defective.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Diagonalization?
Write A = PDP⁻¹ with D diagonal. It is a topic at the expert level in Linear Algebra.
What is the formula for Diagonalization?
The key formula is A = PDP⁻¹.
Where is Diagonalization used in real life?
Real-world examples: Simplifying repeated calculations, such as applying a matrix many times in population models.
What should I learn before Diagonalization?
Make sure you are comfortable with Eigenspaces, the previous topic in Linear Algebra.
What is a common mistake with Diagonalization?
Assuming all matrices are diagonalizable; some are defective.
How can I practise Diagonalization?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Diagonalization?
Move on to Powers of Matrices next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.