1. Introduction to Matlab  Matlab is a software package for technical computation.  Matlab allows you to solve many numerical problems including - arrays and matrix operations - data analysis and graphics - numerical integration - statistical analysis - etc.

2. Today’s Matlab Session  Matlab User Interface Environment   Arrays and Matrix Operations  Plotting Graphs  - X-Y Plots  M-files

3. Running Matlab  Default Matlab Opening Window looks like this:

4. Opening Window  Matlab Opening Window Command window Current (File) Directory Command History Workspace window File Editor

5. Opening Window  Matlab Opening Window  Command Window allows you to enter Matlab commands.  Current Directory Window saves files and data for use.

6. Opening Window  Matlab Opening Window  Command Window allows you to enter Matlab commands.  Current Directory Window saves files and data for use.  Command History Window records the Matlab commands you issued.

7. Opening Window  Matlab Opening Window  Command Window allows you to enter Matlab commands.  Current Directory Window saves files and data for use.  Command History Window records the Matlab commands you issued.  Workspace Window keeps track of the variables you have defined as you execute them in the command window. double (x) returns double precision value for x.

8. Opening Window  Matlab Opening Window  Command Window allows you to enter Matlab commands.  Current Directory Window saves files and data for use.  Command History Window records the Matlab commands you issued.  Workspace Window keeps track of the variables you have defined as you execute them in the command window.  File Edit Window allows you to open existing files including M-files and edit them, create a new file, and save workspace.

9. Desktop icon  Click Desktop icon to modify display window, Maximize Workspace, add/delete windows.

10. Desktop icon  Click Desktop icon to modify display window.  Note: Clicking Desktop to Desktop Layout to Default will allow you to get back to the default layout anytime.

11. Today’s Matlab Session  Matlab User Interface Environment  Arrays and Matrix Operations   Plotting Graphs  - X-Y Plots  M-files

12. Built-In Matrices in Matlab  Square matrix  Equal number of rows and columns. Ex.  >>A=[1,2,3;2,3,4;3,4,5];  Zero matrix  >>A = zeros(3,3)

13. Matrix Addition/Subtraction/Multiplication  Matrix addition/subtraction is done by adding/subtracting element by element  e.g., Adding two 2x2 matrices are done as follows: |a 11 +b 11 a 12 +b 12 |  A + B = |a 21 +b 21 a 22 +b 22 | >> A + B

14. Matrix Addition/Subtraction/Multiplication |a 11 +b 11 a 12 +b 12 |  A + B = |a 21 +b 21 a 22 +b 22 | Ex.

15. Matrix Multiplication  Two matrices, A and B, can be multiplied if and only if the number of columns in A is equal to the number of rows in B  These matrices are then called conformable  For example, (2x3) times (3x2) = 2x2 matrix Note: Matrix multiplication results in an array where each element is a dot product.

16. Matrix Multiplication  Two matrices, A and B, can be multiplied if and only if the number of columns in A is equal to the number of rows in B:

17. Matrix Transpose  Transpose of a matrix (A T or A ’ in Matlab)  The transpose of a matrix is a new matrix where the rows of the original matrix form the columns of the new matrix  Ex. 1 2 3  A= 4 5 6 7 8 9  >>A’  Ans = 1 4 7 2 5 8 3 6 9

18. Matrix Power/Determinant/Inverse  Matrix Power (or Exponentiation)  A 2 = A*A, A 3 = A 2 * A = A*A*A, etc.  >>A^2  >>A^3  etc.

19. Determinant  A determinant of a matrix is a scalar computed from the elements of the matrix  For a 2 X 2 matrix, the determinant of A (or det(A) ) is  |A| = a 11 * a 22 - a 12 * a 21 Ex. >>A=[3,2;5,1] >>det(A) ans = -7

20. Determinant  A determinant of a matrix is a scalar computed from the elements of the matrix  For a 2 X 2 matrix, the determinant of A (or det(A)) is  |A| = a11 a22 - a12 a21  For a 3 X 3 matrix, the determinant of A is |A| = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 - a31 a22 a13 - a32 a23 a11 - a33 a21 a12 etc.

21. Matrix singularity  The inverse of a square matrix, if exists, was defined to be a square matrix such that A*A -1 = I where I is the identity matrix.  Def. A square matrix is singular if and only if (iff) its determinant is zero, i.e. det(A) = 0.  => A square matrix A has an inverse iff det(A) 0. (An important condition to solve simultaneous equations)

22. Ex. Solving Simultaneous Equations  A square matrix A has an inverse iff det(A) 0. Ex. Then C*x=r, where x= => Solution: x=C’*r iff det(C) 0.

23. Ex. Solving Simultaneous Equations  A square matrix A has an inverse iff det(A) 0. Ex. Then C*x=r, where x= => x=C’*r iff det(C) 0. Matlab codes: >>C=[3 2 4;2 5 3;7 2 2] >>CI=inv(C) >>r=[5; 17; 11] >>x=CI*r or >>x=C’*r

24. Today’s Matlab Session  Matlab User Interface Environment  Arrays and Matrix Operations  Plotting Graphs   - X-Y Plots  M-files

25. Display Format in Matlab  Matlab uses a default format that shows four decimal digits. Ex. >>A = 51.1 returns A = 51.1000

26. Matlab ‘plot’ Command Example:  >>x=[1 2 3 4];  >>y=[10 20 20 40]  >>plot (x,y,’o’) %This creates an x-y scatter plot with an ‘o’ marking each point

27. X-Y Plots  >>x=[1 2 3 4];  >>y=[10 20 25 35]  >>plot (x,y,’o’) %This creates an x-y scatter plot with an ‘o’ marking each point  >>grid on % This will add a grid over the plot  >>grid off % This will remove a grid.

28. X-Y Plots  >>x=[1 2 3 4]  >>y=[10 20 25 35]  >>plot (x,y,’o’) This does an x-y scatter plot with an ‘o’ marking each point  If >>plot (x,y) has been used, Matlab would have plotted y vs. x in a line plot.

29. X-Y Plots Another Example:  >>x = sin(0:0.1:10);  >>y = cos(0:0.1:10);  >>plot (x,y) Starting point End point Increment

30. Line Types for X-Y Plots  Use the following commands for customized plots: - solid line -- dashed line -. dash-dot line

31. Line Types for X-Y Plots  Use the following commands for customized plots: - solid line -- dashed line -. dash-dot line Example:  >>x=[1 2 3 4]  >>y=[10 20 30 40]  >>plot (x,y,’--’) %This plots an x-y plot with a dashed line  >>plot(x,y,’-.’) %This plots an x-y plot with a dash-dot line

32. Multiple Plots  Use the following commands for customized plots: - solid line; -- dashed line; -. Dash-dot line Example:  >>x=[0:0.1:5];  >>y1=sin(x);  >>y2=cos(x);  >>plot(x,y1,’--’);  >>hold on %This will keep the plot of x vs. y1

33. Multiple Plots  Use the following commands for customized plots: - solid line; -- dashed line; -. Dash-dot line Example:  >>x=[0:0.1:5];  >>y1=sin(x);  >>y2=cos(x);  >>plot(x,y1,’--’);  >>hold on %This will retain the first plot  >>plot(x,y2,’-.’) % This will plot a cosine curve wit a dash-dot line  >>grid on

34. More Examples of X-Y Plots Example  >>y=sin(0:0.4 : pi*4);  >>plot (x,y) Note: You may enter y=sin(0 : 0.4 : 4*pi) 

35. Plotting Graphs in Matlab Title and Label (exercise!)  Below is a list of commands you will need for labeling graphs in Matlab.  >>title (‘whatever you want’) % Titles the graph  >>xlabel (‘the x-axis label’) % Labels the x-axis  >>ylabel (‘the y-axis label’) % Labels the y-axis

36. Matlab Session  Matlab User Interface Environment  Arrays and Matrix Operations  Plotting Graphs  - X-Y Plots  M-files 

37. M-files  M-files allow you to store programming codes for later use. M-files are ASCII text files similar to C or FORTRAN source codes.  They are called M-files since the filename has the form filename.m.  M-files can be created from the Edit Window (File  New  M-file)

38. Creating an M-file  M-files allow you to store programming codes for later use. M- files are ASCII text files similar to C or FORTRAN source codes.  They are called M-files since the filename has the form filename.m.  Example: First enter the following command sequence in the File Editor Window: File->New->M- file to open the M-file Editor: Ex. First, enter following commands into the M-file Window: x=[0:0.1:4]; y1=sin(x); y2=cos(x); plot(x,y1) hold on plot(x,y2,’o’)

39. Saving an M-file Then, save the script as sin_cos_plot.m: Go to File  Save Workspace as -> name the file as sin_cos_plot.m and click save: x=[0:0.1:5]; y1=sin(x); y2=cos(x); plot(x,y1,’- -’) hold on plot(x,y2,’o’)

40. Editing M-files  They are called M-files since the filename has the form filename.m.  Example: sin_cos_plot.m  We can save the above M-file using the Save and Run icon in the File Editor Window as follows:  1. Save as sin_cos_plot.m.  2. Click on the Save and Run icon.

41. Ex. Plotdata.m  % This is an M-file (or a script) to plot the function  % y=sin(a*t)  %  t=[0:0.1:1];  a=2;  y=sin(a*t)  plot(t,y)  xlabel(‘Time (sec)’)  ylabel(‘y(t)=sin(2t)’)  grid on   Let’s spend 4-5 minutes to work on this, i.e. open File Editor and go to M-File. Type the commands on the left and click on the Save and Run icon. Name the file and run.

42. Ex. plotdata.m  % This is an M-file (or a script) to plot the function  % y=sin(a*t)  %  t=[0:0.1:1];  a=2;  y=sin(a*t)  plot(t,y)  xlabel(‘Time (sec)’)  ylabel(‘y(t)=sin(2t)’)  grid on   Let’s spend 4-5 minutes to work on this.  Let’s run the m-file: >> plotdata

43. Any Questions?