Lecture 07: Matrix Operations

[Music] Hello friends welcome to the course multivariate procedures with r you can recall that in the last couple of lectures we were talking about different types of calculations in the our software and you have understood that the way R does different type of mathematical procedures like as addition subtraction division multiplication that is possible with the scalar as well as data vector and this operations are little bit different than other softwares but I'm sure that now you have understood those operations so now in this lecture and in the next lecture we are going to talk about the Matrix Theory the first question comes here why should we learn this Matrix Theory you see what is the basic objective of this course this is to have the multivariate procedures multivariate means there are more than one variables so whenever we are trying to deal with multivariate procedures be it multivariate normal distribution be it multiple linear regression analysis is be it a discriminant analysis principal component analysis or anything the data is available on more than one variables what do you mean by variables Suppose there are two variables age and weight so there will be one person we try to record her or his age as as well as weight now the second person comes and we try to record uh the weight and then age Etc and so on and similarly you can think that there are more than one variables or more than two variables which are going to affect the process so every set of observation will have not only one value but more than one values suppose if you want to record the data on the health of say 20% and now you define that health is going to be measured by say age weight height blood pressure blood sugar level Etc then each of the 20 persons will have observations on all these variables so the question comes here how are you going to handle such data sets the first question comes we have to give the input to a software and in most of the cases you will see that such observations are arranged in the form of vectors and matrices uh remember I'm not talking of the data Vector I'm talking of the vectors in The Matrix Theory like as row vector or column vectors so it is important for us to understand that how R is handling the Matrix and its operations the before going forward I assume that you all are aware about the Matrix Theory and its various operations and you will agree with me that uh many of the operations in The Matrix Theory they are very much different than the usual escalar operations and when we try to apply those procedures in the r software one aspect ECT which you always have to keep in mind is that R is going to follow the same mathematical rules for the Matrix theory in case if you make a mistake in giving the input to the data or somewhere else are may not be able to exactly diagnose what is the mistake what is the problem but this is only you have to uh see and uh find out what is the problem for example in case if you are trying to multiply two matrices so whenever you are trying to multiply two matrices the rule is that if there are two matrices say A and B and suppose one's order is M byn then the order of the second Matrix has to be n by P or something like this so such type of operations I am assuming that you know so in this uh lecture I am going to introduce this uh matric Theory to brush up your knowledge so that you are aware and then you have a quick review of the basic concepts in The Matrix Theory although there are many many operations in The Matrix Theory and uh it is practically impossible for me to cover all those operation but I will try to take up some standard operations in this lecture and in the next lecture and hopefully after that uh you will be confident that if you want to use any other Matrix procedure you can use it and and you can find out the way or the syntax or the command in the our software so with this objective let us begin our lecture and try to have some glimpse of the Matrix operations in our software right so in this lecture first we try to Define define that what are matrices and as I said that matrices are important object in any calculation and particularly you are trying to do the multivariate analysis then this Matrix Theory becomes an Inseparable part of the whole analysis what is a Matrix Matrix is the arrangement of elements in rows and columns for example if I say in mathematics we try to write down a matrix like this and then I will say here this is the first element a11 this is the a second element A1 two and similarly in the next element a21 a22 and so on so these are called rows and these are called columns right and this subscript if you try to see what is this a11 this is indicating the number row number and column number similarly if I say here a12 so a12 is indicating that is the element in the first row and second column right and if there are such here one 2 up to here P rows and suppose here one two up to here Q columns or say n columns then this is called as a matrix of order P by n right so this Matrix is a rectangular array with P rows and and columns and as we write in the book as I explained you suppose if I say if I try to denote the matrix by here x then usually by xig J we write down its elements but this xig J is written in the book that is how we try to understand the Matrix Theory from the book but in the software and in the programming language usually we try to write this thing as X inside the brackets I comma J so here I goes from here 1 to P and J goes from here 1 to n right now when we are trying to deal with the Matrix operation in any software or in any programming language like as R then the element of this Matrix can be an object for example that can be a number that can be a string and but in mathematics when we are trying to do the mathematical operation then usually we are interested in the numerical Matrix numerical metrix means all the elements in the metric should be some numbers they should not be character otherwise we cannot do the mathematical operations although in order to do the data analysis many times we try to arrange the characters also in the format of a matrix but anyway at the moment we are not going to consider uh those things and we are going to assume that the elements in my Matrix they are the real numbers right so now uh I have two options here that first I try to give you the basic command of the Matrix Theory and then I try to explain you how you can do it in the r software or let me take here some examples and through those examples I will try to explain you that how you can make different operations in The Matrix Theory in this art software so let me try to take here first example suppose I want to create a matrix creating a matrix mean the data has to be arranged in The Matrix now when we are trying to create a matrix what do we need the first ingredient is that we want to know what is the number of rows then we want to know what is the number of columns and then we want to know what is the data that means what are the entries in The Matrix so suppose if I want to create a 4x2 Matrix that means four rows and two columns so if you try to see if there are four rows and two columns then how many data entries you require this will be 4 into 2 that is 8 so now I have to supply this information to the r software so in order to create a metrix or to inform from the r software or to request the r software to create a matrix the command is Matrix M A Rix and then inside the parenthesis you have to give different type of options or information so I want to inform the r software that there are going to be four rows so this is optained by writing n row which is indicating the number of rows and I write here number of rows that is n row equal to four and similarly here n call n call means here number of columns so n call is equal to two so that is indicating that there are four rows and two columns now I want to inform our that what is the numbers or what are the real numbers which are going to take place inside the Matrix so I try to give here eight values which I have combine here in a data Vector so and I give it and I use here this option data so data will communicate to the r software that these are the values which are going to be inside the Matrix so once you execute this thing you will get here this type of outcome X will look like this see here the if you try to see the data I have given here for the sake of understanding in uh an increasing value 1 2 3 4 5 6 7 8 now if you try to see that how this data is going to be entered in the Matrix X it is here one comes here first then comes here two then three then four and from here it goes to the next column five then six then 7 then 8 so if you try to see the data here is arranged column wise and the same operation when I try to do in the r software comes out to be here like this that you can verify this is the same output which is here and what you can observe here I will explain you later that what are these indicating for example this here one this here one and if you try to observe how it is written this is written like first uh You observe these brackets then it is here the number one and then here comma whereas in the column case please try to concentrate where I am writing it is here bracket then there is blank space comma and then it is here one right so if you try to see this is indicating First Column and similarly this is indicating second column similarly this is indicating first row and this is indicating second row and so on right and in case if you want to write what is the address of this number here two so this you can see here this is on the second row and First Column so this is this is how we try to give the or the r is trying to give the addresses to different locations to rows columns as well as individual number so this is I just want to add your attention so that later on when I I'm trying to explain you that how to access any particular row column or element this is going to be helpful right so now you can see here that this here this parameter n row that is defining the number of rows in a metrix n call is trying to define the column numbers of the Matrix and data is assigning the specified value to The Matrix right and when you are not giving any option then this Matrix is trying to assign the observation column wise you can see here and if there are more columns it will go like this right so this is what you have to keep in mind right so now in case if you want to access any particular element of this uh Matrix because it may be required in various multivariate procedures that you want to do something on a particular element and then you need to write a command so that a particular element can be obtained so as I explained you uh just now uh suppose you want to extract this number seven so now if you try to see here this number seven is lying on the third row and in the second column so what you have to write here the name of the Matrix X and using the square brackets you have to write 3 comma 2 and in case if you try to write down here x inside the square bracket 3 comma 2 you will get here the value here 7 right uh one thing I can explain you that as in mathematics we have different types of bracket like as parenthesis curly brackets and square brackets so in R when we are trying to do the mathematical operation then we use only the parenthesis like this the curly brackets and square brackets they are used for some other assignments for example they can have different role for example here you can see when we are trying to find a particular value in The Matrix then we are trying to use this square brackets all right so now in case if my objective is that I want to arrange the same data rowwise vwise as I explained you here that if you don't give here any option then this data is going to be arranged columnwise now I want to arrange it rowwise so how to do it so I will use the same command here same data set but I want to arrange in the rowwise so so I am using the command here Matrix n row and call and data but now I add here one more parameter by row by y r o w right and I write here B is equal to True T Ru in all in uppercase alphabets actually here this true and false false they are The Logical operators and instead of true and false you can also use here say t or F also the first alphabet so if you try to understand what I'm trying to inform to the r software I'm saying by is equal to true that means I am asking r that when you try to arrange the data then you have to arrange the data by rows or and this statement is true that means you please arrange the data row-wise in case if I write down here false by row is equal to false that means I'm trying to inform the r software that you need not to arrange the data rowwise that means obviously the opposite will be correct that the r has to the data column wise so if you try to see here by using this new parameter by row equal to True how this data is going to be obtained so you can see here this Matrix I am calling here as a y and the Y is coming out to be like this you can see here now the same data is here 1 2 two it is going first in the Row first row then it come to the second row 3 to 4 then 5 to 6 then 7 to8 so this is how we try to control this data uh data Arrangement and you can see here that in this uh screenshot I can show you that here I am not using here by byro option where whereas in the second case I'm trying to use here the brro equal to true and the data is here the same 1 to8 numbers number of rows and number of column colums are also the same but here the data is arranged column wise but in the second case the data is arranged rowwise right so let me try to show you first these examples on the our software so that you can be confident about them so let me try to just uh copy this thing so and uh then I try to copy this thing right so even if if you want to write it in the same line that also can be done that is the same thing all right now if you try to see what is here x this is the same operation which I just explained you right now in case if I try to change this command and I try to write down here by row is equal to true so okay I will just write down here as a t only so that you can be confident that t and capital t r u e both are working and let me give this Matrix a name y so you can see here Y is now here like this so that is the same operation which I just uh explained you right and suppose if I want to extract any particular element of here x suppose I want to extract here X3 comma 2 so so this is going to be here 7 whereas if I try to arrange or retrieve the same element in the Y Matrix then it is going to be here six right and that you can see here this is the element in the X Matrix and this is the element in the Y Matrix right so now you you can be confident that yeah this is working without any problem and uh if you try to uh understand here one concept that uh try to look at this screenshot and uh in this screenshot I am not writing anything about by Row in the first uh here say by row so what is the value of your by row so obviously by row here is actually false but this is the default option that even if you don't write anything about by row then R is going to assume it to be false that is the default option so you have to be careful according to your requirement and need you need to use this operation right okay so now we move forward now I would like to introduce you that uh sometimes the data is coming from external sources and it is arranged in some Metric so in this example what you have done here here you have tried to create a matrix of order 4x2 but suppose some unknown data set is coming from outside and you don't know what is the number of rows what is the number of columns Etc and in order to move further you need to have such type of information so the question is that how would you know what are the dimension of the metrix what are the number of rows and what are the the number of columns so if you want to find out the dimension of the Matrix then the command here is diim dim which is the short form of Dimension if you simply write dim and inside the parenthesis X the name of the Matrix you will get here 4x2 which is 4 comma 2 so you have to observe how it is written right and if you want to know the number of rules in The Matrix X then you have to use the command n row inside the parenthesis X right and uh it will give you the option here the answer here here as a four and similarly if you want to know the number of columns then you have to use the command and call X and it will give you the number here to right so this is how you can have different type of information and uh I would like to inform you that you see I'm trying to uh consider here only very limited number of options which are more popular or they are going to be used in our course particularly but but there are many more options so I would recommend you that you please try to look into the help and you simply say help inside the parenthesis within double quotes Matrix and you will get here all the details about the Matrix and there are many other options which are mean avable like as Matrix is Matrix suppose that means if if there is some unknown data set and you want to know whether it can be treated as a matrix or not then this s do Matrix is going to help you but definitely I'm not going into these details and uh but I would certainly like to show you the use of this uh dim command and nro N X right so let me try to clear the screen and let me try to have this metrix here x so you can see here this is here like this and now if I try to write down here DM of X which is here like this 42 that means there are four number of rows and two columns and if I say here end row X so so this is here four and if I say here n call X this is here two and similarly if you try to mean take any other Matrix then the similar observations can be can be obtained for example if I try to show you that if you try to make any mistake then what will happen suppose if I am in the Y Matrix suppose if I try to add here say two more numbers 9 and 10 so there are to Total 10 values but you want to arrange them in the four rows and two columns which is wrong but it will try to say that it is y is giving here like this but it is giving you a warning message and if you try to see what it is trying to do it is trying to take only this eight values and now similarly if you try to make another mistake that if you try to make here the number of rows to be here six that means you need here 12 observation but now you have given only here 10 observation then what happens it is again trying to give you the one war that there is something is wrong you try to find out and if you try to see here how it is doing so it is here 1 2 8 it is arrang say here 10 it is arranging in the five rows and two columns but after that it is repeating so this is you have to keep in mind that this is how the r is working so if you don't give the proper input and if you ignore the warning messages in your simulation or in your computation then the r is going to use this value but you wanted to use only up to here 10 right so this is what you have to be careful so let me try to clear the screen and come back to our slides right so now I try to give you here different types of examples in which we are going to give you the idea that how different types of matrices can be can be defined well these examples will help you when you are working in the multivariate procedure and if you want to input a particular type of data or if you want to do a particular type of operation suppose you want to have a matrix in which all the element should be the same right suppose I want to create a 4x2 Matrix in which all the element should only be five so I can use here the same command Matrix then n row equal to 4 n call equal to 2 but I will say only data equal to here five and as soon as you try to do it here you will get here this type of observation this type of Matrix where you can see there all the eight elements in The Matrix X they are the same F so this type of operations are many times needed right and uh suppose if you want to create a diagonal matrix for example identity Matrix is a diagonal matrix in which all the elements on the diagonal elements are one and of diagonal elements are zero what are the diagonal of diagonal elements if you try to take a square Matrix then uh if you try to write down here elements like a12 and so on A2 1 a21 a22 Etc like this then this is actually here the diagonal so these are called the diagonal elements and all these elements which are not on the diagonal they are called as of diagonal elements and this part is called as upper diagonal and the elements in the lower part they are called as lower diagonal right so if you want to create a diagonal matrix suppose you want to create a identity Matrix of order 2x two then the command here is e i a and you see here one which is the value of data and uh you define the number of rows and number of columns as n row equal to two and N call equal to two and if you try to see here this outcome will come here like this right and if you want to have any other value on the diagonal then also you can play with this value one and I will try to show you it on the r console right and uh uh there is another Concept in The Matrix Theory which is transpose of a matrix transpose of a matrix is defined s when you try to Interchange the rows and columns suppose if I say here my Matrix here is 1 2 3 4 and now I try to Define here the transpose which is dened by X Prime which is here x transpose or then the first row will become here First Column and the second row will become here second column so this is a very popular Operation and if you want to do it then the command here is TX but let me try to show you with the an example suppose I try to Define here the uh the same Matrix X which is 4x2 Matrix and the data is here 1 to8 right this is another way of writing the elements 1 2 3 4 up to 8 like this right and by row is equal to true so you get here this Matrix right 1 2 3 4 5 6 7 8 like this in which the data is arranged rwise now we want to find out the transpose of this Matrix the r command to find out the transpose of a matrix is T inside the parenthesis you have to write down the Matrix so I write down here T inside the parenthesis X and then you can see here the outcome and suppose I stored this transpose in a and Matrix XT so this is the here the outcome that now this one and two they are coming on the First Column three and four they are coming on the second column five and six they are coming coming on the third column and seven and eight they are coming on the fourth column and if you try to is compare them in this screenshot you can see here your X Matrix was like this one two were the first row but now one two are becoming the First Column the second row is becoming here the second column third row is becoming the third column fourth row is becoming the fourth column right so this is how we try to do but before going forward let me try to uh show you these operations on the r console so that you feel more comfortable suppose I want to create here a matrix 4X two in which all the elements are going to be here five so you can see here five and similarly if you want to change this number suppose I want to give this number to 15 so you can see here x is coming out to here 15 right and uh similarly if you want to create here a diagonal matrix so you can see here this is here your diagonal matrix of order 2x2 which is an identity Matrix and if you want to change the order of this Matrix that means if you want to h a 3X3 identity Matrix now you can see here there are three elements and if you try to change the value on the diagonal suppose if I want to make it here two so you can see here now this is a diagonal matrix uh of in which all the diagonal elements are two and on the other hand I can also show you that if you want to make diagonal matrix of different numbers so I can see you here see uh 1 comma 2 comma 3 and you will see here this will come out to be here like this that the first diagonal has value one second diagonal two and third diagonal here three right right and uh before I try to give you other thing let me first try to show you the transpose concept and then I will try to show you something where you have to be careful so this is your here x Matrix and if you want to find out your transpose of X this is coming here like this right okay so now if you try to see suppose if I try to write down this command here which I used earlier right right if you try to recall this thing X this is your here x now suppose if I don't write this parameters here and suppose if I write down here four I delete n row I delete the word n call and I delete the word data and if I try to execute it it will give me the same value no it is not happening because the first value it has taken which is here the data right so you have to be very careful that you should not go for a shortcut but I always recommend that you try to write down this n row end call all these values very carefully and if you for example if you try to see here if I try to write down here data equal to this then what happens you can see here it is giving you the same metrix but in this case what is happening it is trying to take here the first value four as the number of rows and second value as a number of call number of columns and if I try to suppose change it you can see here this becomes here like this but as a lay man it is very difficult for me to understand by looking as 4A 2 and 2A 4 that uh whether it is the number of rows first then the number of column or vice versa so my simple suggestion to you all is that always write the parameter name so that it is 100% clear to everyone what we are trying to do and there is absolutely no confusion okay so now uh we come to an end to this uh lecture and you can see here uh I have taken very simple topics and my objective was to introduce you and make you familiar with the metrics in our software that how are you going to handle it and you can see here it is not difficult but as I shown you in the last uh example that you have to Be watchful you have to be careful when you are trying to define the values in the metrix options uh for example the values of different parameters you have to Define it carefully and please write write all the commands always in detail so that after some time if you are looking at the same command at least you can recall them beside this thing it is very important R will do all Matrix operation following the rules of Matrix and this is only you who has to give the input in the right format so this is important for you to understand the basic functionality of the Matrix Theory basic mathematical rules in the math uh in The Matrix Theory and then try to write down the syntax and commands accordingly so I would request you all that you try to take some simple example and try to practice that how you can Define different types of matrices try to play with the number of row number of columns diagonal matrix Etc and try to see that how you can do it and make yourself familiar with the Matrix Theory and in the next lecture I will try to take some more basic operations in The Matrix Theory like as addition subtraction inverse Etc so I would request you that you also try to revise those concept from your book on The Matrix theory that how you use to obtain them so you try to practice it and I will see you in the next lecture till then [Music] goodbye [Music] [Music] [Music]