1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 5: State Deviations and Fundamentals of Linear Algebra

2. University of Colorado Boulder  Homework 2– Due September 12  Make-up Lecture ◦ Today @ 3pm, here 2

3. University of Colorado Boulder  Effects of State Deviations  Linear Algebra (Appendix B) 3

4. University of Colorado Boulder 4 Quantifying Effects of Orbit State Deviations

5. University of Colorado Boulder  Quantification of such effects is fundamental to the OD methods discussed in this course! 5 Time

6. University of Colorado Boulder  Let’s think about the effects of small variations in coordinates, and how these impact future states. 6 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Example: Propagating a state in the presence of NO forces

7. University of Colorado Boulder  What happens if we perturb the value of x0? 7 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0)

8. University of Colorado Boulder  What happens if we perturb the value of x0? 8 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf+Δx, yf, zf, vxf, vyf, vzf)

9. University of Colorado Boulder  What happens if we perturb the position? 9 Initial State: (x0, y0, z0, vx0, vy0, vz0) Force model: 0 Initial State: (x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0) Final State: (xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf)

10. University of Colorado Boulder  What happens if we perturb the value of vx0? 10 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0, y0, z0, vx0-Δvx, vy0, vz0)

11. University of Colorado Boulder  What happens if we perturb the value of vx0? 11 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Final State: (xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf) Initial State: (x0, y0, z0, vx0+Δvx, vy0, vz0)

12. University of Colorado Boulder  What happens if we perturb the position and velocity? 12 Force model: 0

13. University of Colorado Boulder  We could have arrived at this easily enough from the equations of motion. 13 Force model: 0

14. University of Colorado Boulder  This becomes more challenging with nonlinear dynamics 14 Force model: two-body

15. University of Colorado Boulder  This becomes more challenging with nonlinear dynamics 15 Force model: two-body Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

16. University of Colorado Boulder  This becomes more challenging with nonlinear dynamics 16 Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body

17. University of Colorado Boulder 17 Select Topics in Linear Algebra

18. University of Colorado Boulder  Matrix A is comprised of elements a i,j  The matrix transpose swaps the indices 18

19. University of Colorado Boulder  Matrix inverse A -1 is the matrix such that  For the inverse to exist, A must be square  We will treat vectors as n×1 matrices 19

20. University of Colorado Boulder 20

21. University of Colorado Boulder 21

22. University of Colorado Boulder  If we have a 2x2, nonsingular matrix: 22 Asking you to invert a full 2x2 matrix on an exam is fair game!

23. University of Colorado Boulder  The square matrix determinant, |A|, describes if a solution to a linear system exists: 23  It also describes the change in area/volume/etc. due to a linear operation:

24. University of Colorado Boulder 24

25. University of Colorado Boulder  A set of vectors are linearly independent if none of them can be expressed as a linear combination of other vectors in the set ◦ In other words, no scalars α i exist such that for some vector v j in the set {v i }, i=1,…,n, 25

26. University of Colorado Boulder  The matrix column rank is the number of linearly independent columns of a matrix  The matrix row rank is the number of linearly independent rows of a matrix  rank(A) = min( col. rank of A, row rank of A) 26

27. University of Colorado Boulder 27

28. University of Colorado Boulder 28

29. University of Colorado Boulder  When differentiating a scalar function w.r.t. a vector: 29

30. University of Colorado Boulder  When differentiating a function with vector output w.r.t. a vector: 30

31. University of Colorado Boulder  If A and B are n×1 vectors that are functions of X: 31

32. University of Colorado Boulder  The n×n matrix A is positive definite if and only if: 32  The n×n matrix A is positive semi-definite if and only if:  

33. University of Colorado Boulder  The point x is a minimum if 33 and is positive definite.

34. University of Colorado Boulder  Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where 34  

35. University of Colorado Boulder  Other identities/definitions in Appendix B of the book ◦ Matrix Trace ◦ Maximum/Minimum Properties ◦ Matrix Inversion Theorems  Review the appendix and make sure you understand the material 35