MATLAB Tutorial You need a small numb er of basic commands to start using MATLAB This short tutorial describ es those fundamental commands You need to create vectors and matrices to change operate with them Those are all short highlevel commands b ecause MATLAB them and to constantly works with matrices I b elieve that you will like the p ower that this software gives by a series of short instructions to do linear algebra create E create u change E multiply E u E eye E E u v E u 2 3 2 3 2 3 2 3 4 5 4 5 4 5 4 5 The word eye stands for the identity matrix The submatrix u E picks out column E resets the entry to The command E u multiplies the The instruction u All these commands are rep eated in our list b elow Here is an example matrices E and inverting a matrix and solving a linear system of create A create b invert A solve Ax b A C invA x Anb or A ones eye b x C b 2 3 2 3 2 3 2 3 4 5 4 5 4 5 4 5 The matrix of all ones was added to eye and b is its third column Then invA pro duces inverse matrix normally in decimals for fractions use format rat The system Ax b the solved by x invA b which is the slow way The backslash command x Anb uses is never computes the inverse matrix When the right Gaussian elimination if A is square and 0 A the solution x must b e The transpose symbol side b equals the third column of vector Then A x picks out the third column of A and we have Ax b makes x a column Here are a few comments The comment symb ol is The symb ols a and A are dierent MATLAB is casesensitive Typ e help slash for a description of how to use the backslash symb ol The word help b e followed by a MATLAB symb ol or command name or Mle name can Note The command name is upp er case in the description given by help but must b e lower case in actual use And the backslash Anb is dierent when A is not square To display all digits typ e format long The normal format short gives digits after the decimal after a command avoids display of the result A semicolon display the identity matrix A ones will not uparrow cursor to return to previous commands Use the How to input a row or column vector u has one row with three comp onents a matrix rows separated by semicolons a matrix v has three 0 0 v or v u transposes u to pro duce the same v w generates the row vector w with unit steps takes steps of to give u u How to input a matrix a row at a time A has two rows always a semicolon b etween rows A also pro duces the matrix A but is harder to typ e 0 T 0 B is the transpose of A Thus A is A in MATLAB How to create sp ecial matrices diagv pro duces the diagonal matrix with vector v on its diagonal to eplitzv gives the symmetric constantdiagonal matrix with v as rst row and rst col umn to eplitzw v gives the constantdiagonal matrix with w as rst column and v as rst row onesn gives an n n matrix of ones zerosn gives an n n matrix of zeros eyen gives the n n identity matrix randn gives an n n matrix with random entries b etween and uniform distribution randnn gives an n n matrix with normally distributed entries mean and variance onesm n zerosm n randm n give m n matrices onessizeA zerossizeA eyesizeA give matrices of the same shap e as A How to change entries in a given matrix A A resets the entry to equal A v resets the third row to equal v A w resets the second column to equal w The colon symb ol stands for al l all columns or all rows A A exchanges rows and of A How to create submatrices of an m � n matrix A Ai j returns the i j entry of the matrix A scalar matrix Ai returns the ith row of A as row vector A j returns the j th column of A as column vector A returns rows from to and columns from to as matrix A returns rows and and all columns as n matrix A returns one long column formed from the columns of A mn matrix triuA sets all entries b elow the main diagonal to zero upp er triangular trilA sets all entries ab ove the main diagonal to zero lower triangular Matrix multiplication and inversion A B gives the matrix pro duct AB if A can multiply B A B gives the entrybyentry pro duct if sizeA sizeB ;1 invA gives A if A is square and invertible pinvA gives the pseudoinverse of A AnB gives inv A B if invA exists backslash is left division Anb gives the solution to Ax b if invA exists x See help slash when A is a rectangular matrix Numb ers and matrices asso ciated with A detA is the determinant if A is a square matrix rankA is the rank numb er of pivots dimension of row space and of column space sizeA is the pair of numb ers m n traceA is the trace sum of diagonal entries sum of eigenvalues nullA is a matrix whose n r columns are an orthogonal basis for the nullspace of A orthA is a matrix whose r columns are an orthogonal basis for the column space of A Examples E eye E creates a elementary elimination matrix A subtracts times row of A from row E A b creates the augmented matrix with b as extra column B eye P E creates a p ermutation matrix E triuA trilA diagdiagA equals A Note that Builtin Mles for matrix factorizations all imp ortant L U P luA gives three matrices with P A LU e eig A is a vector containing the eigenvalues of A S E eig A gives a diagonal eigenvalue matrix E and eigenvector matrix S with AS S E If A is not diagonalizable to o few eigenvectors then S is not invertible Q R qrA gives an m m orthogonal matrix Q and m n triangular R with A QR Creating Mles Mles are text les ending with m which MATLAB uses for functions and scripts A script which may b e executed often and can b e placed in an mle so is a sequence of commands do not have to b e retyp ed MATLABs demos are examples of these scripts the commands house Most of MATLABs functions are actually mles An example is the demo called viewed by writing type xxx where xxx is the name of the function and can b e To write your own scripts or functions you have to create a new text le with any name you like provided it ends with m so MATLAB will recognize it Text les can b e created saved with any text editor like emacs EZ or vi A script le is simply a list of edited and MATLAB commands When the le name is typ ed at the MATLAB prompt the contents of le will b e executed For an mle to b e a function it must start with the word function the followed by the output variables in brackets the function name and the input variables Examples function CmultA rrankA 0 A A C Save the ab ove commands into a text le named multm Then this funtion will take a pro duct C The variable r is not returned b ecause it matrix A and return only the matrix was not included as an output variable The commands are followed by so that they will printed to the MATLAB window every time they are executed It is useful when not b e dealing with large matrices Here is another example function VDrpropertiesA rank eigenvalues and eigenvectors of A This function nds the mnsizeA if mn VDeigA rrankA else dispError The matrix must be square end Here the function takes the matrix A as input and only returns two matrices and the rank comment The function checks to see if the input ma as output The is used as a trix is square and then nds the rank eigenvalues and eigenvectors of a matrix A Typing propertiesA only returns the rst output V the matrix of eigenvectors You must typ e VDrpropertiesA to get all three outputs Keeping a diary of your work The command diaryle tells MATLAB to record everything done in the MATLAB window and save the results in the text le named le Typing diary on or diary o b e viewed using a text editor or printed using lpr toggles the recording Old diary les can MATLAB they can b e viewed using the typ e le command in unix In Saving your variables and matrices The command diary saves the commands you typ ed as well as MATLABs output but it do es not save the content of your variables and matrices These variables can b e listed by which also lists the sizes of the matrices The command save xxx the command whos save the matrices and all variables listed by the whos command into the le named will xxx MATLAB lab els these les with a mat extension instead of m which are scripts or by MATLAB at a later time by typing load xxx functions xxxmat les can b e read Graphics The simplest command is plotx y which uses two vectors x and y of the same length The will ints x y b e plotted and connected by solid lines p o i i If no vector x is given MATLAB assumes that xi i Then ploty has equal spacing xaxis the p oints are i y i on the The typ e and color of the line b etween p oints can b e changed by a third argument The with no argument is a solid black line Use help plot for many options we indi default cate only a few 0 0 MATLAB plotx y r plots in r red with for p oints and dotted line 0 0 0 0 MATLAB plotx y is a dashed line and plotx y is a dotted line You can omit the lines and plot only the discrete p oints in dierent ways 0 0 0 0 0 0 0 0 plotx y o gives circles Other options are or or For two graphs on the same axes use plotx y X Y Replace plot by loglog or change one or b oth axes to logarithmic scale The command semilogy or semilogx to axis a b c d will scale the graph to lie in the rectangle a x b c y d To title the lab el the xaxis or the y axis put the desired lab el in quotes as in these examples graph or title height of satellite xlabel time in seconds y label height in meters The command hold keeps the current graph as you plot a new graph Rep eating hold To print or save the graphics window in a le see help print or use will clear the screen print Pprintername print d lename MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.