1. REVIEW OF MATHEMATICS

2. Review of Vectors Analysis GivenMagnitude of vector: Example: Dot product:  is the angle between the two vectors. Example:

3. Review of Vectors Analysis Vectors and are said to be perpendicular or orthogonal if Example: Note that the above vectors represent the unit vectors for the X- axis and Y-axis. They are definitely perpendicular or orthogonal.

4. Cross product:  is the angle between the two vectors. Review of Vectors Analysis Example:

5. Review of Vectors Analysis The cross product of and provides us with a vector which is perpendicular to both and Example: Note that the above vectors represent the unit vectors for the X- axis and Y-axis respectively. Their cross product is the unit vector for the Z-axis, which is definitely perpendicular to both the X-axis and the Y-axis.

6. Review of Vectors Analysis Note that the unit vectors for the right handed Cartesian reference frame are orthonormal basis vectors, i.e.

7. Vector triple product: Review of Vectors Analysis Example:

8. Review of Vectors Analysis Scalar triple product: Example:

9. Review of Vectors Analysis Given Example: where  is a any constant Given Example:

10. Review of Vectors Analysis Given Example:

11. Review of Vectors Analysis Given Example:

12. Review of Vectors Analysis where A is a matrix of dimension comparable to the vector being multiplied Given Example:

13. Eigenvalues and Eigenvectors Let A be an n  n matrix. If there exists a and a nonzero n  1 vector such that then is called an eigenvalue of A and is called an eigenvector of A corresponding to the eigenvalue Let I n be a n  n identity matrix. The eigenvalues of n  n matrix A can be obtained from: A n  n matrix A has at least one and at most “n” distinct eigenvalues

14. Example 1: Eigenvalues and Eigenvectors Find the eigenvalues of Solution:

15. Example 2: Eigenvalues and Eigenvectors What is the eigenvector of at =1?

16. Multiply 3 rd eqn by -5 and add it to 1 st eqn to eliminate Example 2: Eigenvalues and Eigenvectors

17. Divide 2 nd eqn by and simplify using the known result: Example 2: Eigenvalues and Eigenvectors

18. Story so far: We can obtain a normalized eigenvector using:

19. Trigonometric Functions