1. Diagonalization Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Diagonalization
If an nxn matrix has n independent eigenvectors (i.e. enough to span ℝn),
we will be able to perform a “similarity” transformation, to obtain a diagonal
matrix that has the eigenvalues of the original matrix on the diagonal.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Diagonalization
If an nxn matrix has n independent eigenvectors (i.e. enough to span ℝn),
we will be able to perform a “similarity” transformation, to obtain a diagonal
matrix that has the eigenvalues of the original matrix on the diagonal.
Here is the procedure:
Given an nxn matrix A, find all eigenvalues and eigenvectors, then form a matrix with the eigenvectors as columns (we will call this matrix P).
Next find the inverse of P.
Now multiply: P-1AP=D, D is the diagonal matrix.
We can convert back by multiplying: A=PDP-1.
On the following slides are a couple of examples.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Use the shortcut for the inverse of a 2x2 matrix
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Next we multiply P-1AP.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Next we multiply P-1AP.
Notice that matrix AP comes out to be just the eigenvectors times the eigenvalues, as expected.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

12. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Next we multiply P-1AP.
We end up with a diagonal matrix that has the eigenvalues on the diagonal (and 0 everywhere else).
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

13. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

14. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

15. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

16. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

17. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

18. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Use the shortcut for the inverse of a 2x2 matrix
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

19. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Use the shortcut for the inverse of a 2x2 matrix
Next we multiply P-1AP.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

20. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Use the shortcut for the inverse of a 2x2 matrix
Next we multiply P-1AP.
Notice that matrix AP comes out to be just the eigenvectors times the eigenvalues, as expected.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

21. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Use the shortcut for the inverse of a 2x2 matrix
Next we multiply P-1AP.
We end up with a diagonal matrix that has the eigenvalues on the diagonal (and 0 everywhere else).
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

22. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

23. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

24. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

25. Diagonalization
Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A.
Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1.
Next we multiply P-1AP.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

26. Diagonalization 5 𝑥 2 −4𝑥𝑦+2 𝑦 2 =30 x y x’ y’
Here is an example from geometry. The given equation represents an ellipse.
5 𝑥 2 −4𝑥𝑦+2 𝑦 2 =30 x y x’ y’ 𝜃
Notice that the axes of the ellipse are rotated from the standard x and y coordinate axes.
Through our diagonalization process, we will find a more appropriate coordinate system where the new axes, call them x’ and y’, are aligned with the ellipse. This will simplify the equation of the ellipse.
First we have to get this equation into matrix form, so we can use our linear algebra.
5 𝑥 2 −4𝑥𝑦+2 𝑦 2 =30→ 𝑥 𝑦 5 −2 − 𝑥 𝑦 =30
Any ellipse or hyperbola can be written in this format.
We will find eigenvalues and eigenvectors of this matrix to find the new axes that match up with the ellipse. This gives us an alternate basis for ℝ 2 .
We can also find the rotation angle for the new axes.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

27. Diagonalization 5−𝜆 −2 −2 2−𝜆 =0→ 5−𝜆 2−𝜆 −4=0 y x’
5−𝜆 −2 −2 2−𝜆 =0→ 5−𝜆 2−𝜆 −4=0 x y x’ y’ 𝜃
𝜆 2 −7𝜆+6=0→ 𝜆−1 𝜆−6 =0
𝜆=1 𝑜𝑟 𝜆=6
Find eigenvectors for each eigenvalue.
𝜆=1: 4 −2 −2 1 → 𝑥 1 = 𝑥 2 2 → 𝑥 = 1 2
𝜆=6: −1 −2 −2 −4 → 𝑥 1 =−2 𝑥 2 → 𝑥 = −2 1
Notice that these eigenvectors form an orthogonal basis for ℝ 2 .
Our transformation matrix is 𝑃= 1 − and its inverse is 𝑃 −1 = −2 1
The diagonalization process looks like this:
𝐷=𝑃 −1 𝐴𝑃= − −2 − − =
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

28. Diagonalization y’ 𝑥′ 𝑦′ 1 0 0 6 𝑥′ 𝑦′ =30→ 𝑥′ 2 +6 𝑦′ 2 =30 x’
Now that we have the diagonalized matrix, we can see how to write the simplified equation. x’ y’
𝑥′ 𝑦′ 𝑥′ 𝑦′ =30→ 𝑥′ 2 +6 𝑦′ 2 =30
Here is the graph, using the new axes.
To find the rotation angle we need to compare the P matrix to the standard 2x2 rotation matrix.
Our transformation matrix is 𝑃= 1 −2 2 1
The matrix that rotates the plane counterclockwise by angle 𝜃 is 𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
To see how this matches up with our P matrix we need to realize that matrix P does two things – a stretch and a rotation.
Each column vector of P has length 5 , so we could re-write the matrix as
𝑃= − Now we can compare to find that 𝜃= 𝑐𝑜𝑠 − ≈63°.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB