Diagonalization #

Diagonalisation expresses a matrix using its eigenvectors and eigenvalues when possible.

From lecture explanation, diagonalisation is one of the most powerful tools because it converts a complicated matrix into a much simpler form.

Instead of working with a full matrix, we work with a diagonal matrix, which is much easier to analyse and compute.

Key Idea: If a matrix has enough independent eigenvectors, it can be rewritten as a diagonal matrix using a change of basis. This simplifies matrix operations significantly.

Core Idea #

A matrix is diagonalizable if we can write it as:

Where:

• P is a matrix whose columns are eigenvectors
• D is a diagonal matrix containing eigenvalues

Why Diagonalization Works (Lecture Insight) #

From lectures:

• Matrix multiplication represents transformation
• Eigenvectors are directions that do not change direction
• Eigenvalues tell how much scaling happens

So if we express everything in terms of eigenvectors:

• transformation becomes simple scaling
• no mixing of components

Change of Basis Interpretation #

Diagonalization is essentially a change of coordinate system.

Interpretation:

• P converts to eigenvector basis
• A acts as scaling
• P^{-1} converts back

From lecture: this is why diagonal matrices are easy — they scale each coordinate independently.

When is a Matrix Diagonalizable? #

A matrix is diagonalizable if it has enough independent eigenvectors.

Lecture rule:

• Distinct eigenvalues ⇒ automatically diagonalizable
• Repeated eigenvalues ⇒ must check eigenvectors

Algebraic vs Geometric Multiplicity #

From slides:

• Algebraic multiplicity = number of times eigenvalue appears
• Geometric multiplicity = number of independent eigenvectors

Condition:

for diagonalization.

Special Case: Symmetric Matrices #

From lecture:

Symmetric matrices are always diagonalizable.

Even stronger:

Where:

• Q is orthogonal
• eigenvectors are orthonormal

This is called spectral decomposition.

Why Diagonalization is Useful #

From lecture and webinar:

1. Matrix Powers #

Easy because D is diagonal.

2. Matrix Inverse #

3. Understanding Structure #

• Eigenvalues show scaling behaviour
• Eigenvectors show directions

Step-by-Step Method (Exam) #

1. Find eigenvalues using:

2. Find eigenvectors:

3. Form matrix P using eigenvectors

4. Form diagonal matrix D using eigenvalues

5. Verify:

Example #

Given:

Steps:

• Find eigenvalues
• Find eigenvectors
• Construct P and D

Geometric Interpretation #

• Original space → transformed space
• Diagonalization aligns axes with eigenvectors

Result:

• transformation becomes pure scaling

Connection to SVD #

From lecture:

• Diagonalization works for square matrices
• SVD generalises this idea

Both aim to simplify matrix structure.

Hidden Exam Pattern #

From lectures:

• Often combined with:
  • eigenvalues
  • matrix powers
  • rank

Common Mistakes #

• Not checking number of eigenvectors
• Assuming repeated eigenvalues ⇒ diagonalizable
• Incorrect inverse of P
• Mixing order of multiplication

Strategy to Prepare #

1. Practice eigenvalue problems
2. Check independence of eigenvectors
3. Practice forming P and D
4. Solve matrix power problems

Quick Summary Table #

References #

• Lecture slides (Eigenvalues, Decomposition)
• Course handout
• Webinar discussions

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