# Diagonalization

### Diagonalization

#### Lessons

An $n \times n$ matrix $A$ is diagonalizable if and only if $A$ has $n$ linearly independent eigenvectors.

If you have $n$ linearly independent eigenvectors, then you can use the formula

$A=PDP^{-1}$

where:
The columns of $P$ are the eigenvectors
The diagonal entries of $D$ are eigenvalues corresponding to the eigenvectors
$P^{-1}$ is the inverse of $P$

To see if a matrix is diagonalizable, you need to verify two things
1. There are n linearly independent eigenvectors
2. $AP=PD$

Useful fact: If $A$ is an $n \times n$ matrix with $n$ distinct eigenvalues, then it is diagonalizable.
• Introduction
Diagonalization Overview:
a)
The Formula $A=PDP^{-1}$
• Why is it useful?
• Finding High Powers of $A$

b)
How to Diagonalize a Matrix
• Calculate the eigenvalue
• Find the eigenvectors
• Combine the eigenvectors to create $P$
• Use the eigenvalues to create $D$
• Find $P^{-1}$

c)
How to See if a Matrix is Diagonalizable
• Finding the basis of each eigenspace
• Create a Matrix $P$ and Matrix $D$
• Check if $AP=PD$

• 1.
Computing a Matrix of High Power
Let $A=PDP^{-1}$, then compute $A^4$ if

• 2.
Determining if a Matrix is Diagonalizable
Is the following matrix diagonalizable?

• 3.
Is the following matrix diagonalizable?

• 4.
Diagonalizing the Matrix
Diagonalize the following matrix.

• 5.
Proof relating to Diagonalization
Show that if A is diagonalizable and invertible, then so is $A^{-1}$.