1. PHY 301: MATH AND NUM TECH Chapter 5: Linear Algebra Applications I.Homogeneous Linear Equations II.Non-homogeneous equation III.Eigen value problem

2. PHY 301: MATH AND NUM TECH 5.1 Homogeneous linear Equations 5.1. Solution to homogeneous linear equations.

3. PHY 301: MATH AND NUM TECH 5.1 Homogeneous linear Equations 5.1 Solution to homogeneous linear equations cont’d. E5.1-1 Consider the system of equations What are the possible solutions? If applicable, find the equation of the set of solutions.

4. PHY 301: MATH AND NUM TECH II. INHOMOGENEOUS EQN 5.2 Inhomogeneous Equations:

5. PHY 301: MATH AND NUM TECH II. INHOMOGENEOUS EQN 5.2 Inhomogeneous Equations:

6. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM

7. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM

8. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM

9. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM

10. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM

11. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM: Diagonalization of matrices An nxn matrix A is said to be diagonalizable if there exist a matrix P such that A=PDP -1 (or equivalently P -1 AP=D) where D is a diagonal matrix. Theorem: P exists if and only if there exist n independent eigenvectors of A; Consider the n independent eigenvectors of A: in that case P is made up of the eigenvectors of A, entered as column coefficients. D is in addition found to be made up of the corresponding eigenvalues of A Proof for n=3 (general case identical)

12. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM: Diagonalization of matrices Now let’s compute PD where D is the diagonal matrix of the eigenvalues arranged in the same order as their corresponding eigenvectors in the matrix P. We get: But that’s exactly the same answer as we got on the previous page computing AP!!! So we can conclude that: AP=PD and thus, if P is invertible (i.e. its det is non-zero and thus the eigenvector are linearly independent!), we finally obtain: A=PDP -1 where P is the matrix of the eigenvectors of A in column placement and D is the diagonal matrix of the eigenvalues of A is in the same position as the eigenvectors in P: If some of the eigenvalues are degenerate one can carry out the same procedure as long as one can find n linearly independent eigenvectors. In the next chapter we’ll look at a few examples of this procedure in concrete physical cases.

13. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM: Diagonalization of matrices Example of diagonalization of A=

14. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM: Diagonalization of matrices Example of diagonalization of A=

15. PHY 301: MATH AND NUM TECH 5.3 EIGENVALUE PROBLEM 5.3 Eigenvalue PROBLEM E5.3-3 Consider the matrix: Find the characteristic equations, eigenvalues and eigenvectors (normalized). Explain whether or not the matrix can be diagonalized, considering the eigen vectors. If it can, continue and find matrices P and D and verify that P does indeed diagonalize A. E5.3-4 Repeat above steps for and then check with mathematica eigenvalues and eigenvectors. E5.3-5 Consider the matrix of a 30 degree counterclockwise rotation in 2 dimensions. Can you find real eigenvalues/ eigenvectors? Comment on your result as it pertains to the diagonalization of a rotation matrix. E5.3-6 Take an 4x4 matrix of your choice. Compute its eigenvectors and eigenvalues using mathematica. Still using mathematica compute P -1 AP and verify that you indeed get the diagonal matrix of its corresponding eigenvalues. If the matrix yu found is not diagonalizable choose a different one and repeat calculation