Matrix Applications and Computations

_by

_{R. E. White}

_{Department of
Mathematics}

_NCSU

_{Draft Date:2-03-02}

These notes are a possible one-credit course for students with one semester of a calculus. Basic matrix algebra for all undergraduates of the sciences/engineering has for a number of decades migrated among calculus/differential equations. Mathematics departments have struggled with the placement of basic matrix algebra. Also, in recent decades the sciences have used more discrete models, mathematics and computation. During 1999-2001 ABET (Accrediting Board for Engineering and Technology) has phased in recommendations which acknowledge the importance of discrete and numerical methods. Basic matrix algebra now needs a permanent home, and not just as an add-on to the traditional calculus/differential equations sequence.

The following notes should be considered as a working first draft of a possible course for all undergraduate science and engineering students. Most of the proofs will be of little interest to these students, but I did have a nice time trying to make the notes self-contained. Not all the applications need to be done, but the students should be aware of them. So, this leaves the basic matrix operations and their properties, which are formally stated in Propositions 1-8.

Matrices and Visualization

1. Matrices and Plotting Inventory

2. Curves in the Space and Projectiles

3. Surfaces and Optimization

Algebraic Systems of Equations

4. Applications and Matrix-Vector Products

5. Matrix-Matrix Products

6. Gauss Elimination

Inverse Matrices

7. Definition of an Inverse Matrix and Examples

8. Properties of Inverse Matrices

9. Applications to Circuits, Structures and Mixing

Least Squares and Data Fitting

10. Curve Fitting to Data

11. Normal Equations and Polynomial Approximation

12. Multilinear Approximation and Parameter Identification

Eigenvalues and Time Dependent Applications

13. Definition of Eigenvalues and Two-tank Mixing

14. Computation of Eigenvalues and Eigenvectors

15. Application to Three-tank Mixing and Lead Poisoning

Parameter Identification

16. Radioactive Decay

17. Steady State Heat Diffusion

18. Spread of Information

19. SIR Epidemic Model

Matlab m-files

The following Matlab demonstration m-files are associated with the above lectures. They have a number of invocations of the Matlab command pause, which requires the viewer to hit any key to proceed with the execution of the code. All these codes should be viewed with two windows, one for the code and one for the execution. These codes can easily be used to do a number of the homework problems.

1. inventory.m

2. proj3d.m

3. box3d.m

4. matvec.m

5. matmat.m

6. gauss_el.m

7. inv_mat.m

8. blockg_el.m

9a. circuit3.m

9b. tank3.m

9c. bridge1.m

10. oper_transpose.m