WEBVTT 00:13.870 --> 00:18.460 hello and welcome to the day four of the third week of the structure series i hope it 00:18.460 --> 00:23.250 was the green pal and i also hope that you are following the solution that i have 00:23.250 --> 00:28.500 been making like guess you you have practices lot regarding the position if not then 00:28.500 --> 00:37.620 please do it and at this context i will also try to make a note that you can consult 00:37.620 --> 00:41.620 the book by which was suggested originally for this position part and and group multiplication 00:41.620 --> 00:45.770 table formation you can consult any any standard book on group theory any mathematical 00:45.770 --> 00:49.210 book and also you consult the book by so with knowledge that we have gathered so 00:49.210 --> 00:56.289 far we will now utilized them to from something called representation 00:56.289 --> 01:04.580 of a group ok so what what do we understand when you say representation of a group 01:04.580 --> 01:10.530 so we have learnt to right a group in terms of the symmetry operations ok so when we particularly 01:10.530 --> 01:15.030 talk about the symmetry point groups now we will write in terms say all this 01:15.030 --> 01:23.090 so implies notations like e c n sigma v sigma e sigma d i s n l all those things so thats 01:23.090 --> 01:31.120 fine now if you have to utilized the group theory to you know understand you know 01:31.120 --> 01:37.260 several molecular property particularly in the context of the symmetry of the molecule 01:37.260 --> 01:43.700 then we need to have you know proper mathematical representation of this symmetry 01:43.700 --> 01:46.810 operations ok so what what is you know way to form mathematical representation 01:46.810 --> 01:54.290 of the symmetry operations so if you are you know familiar with you know quantum 01:54.290 --> 01:59.680 mechanics then you probably know that one of this one of the ways is to utilize 01:59.680 --> 02:03.540 matrix so here in case of you know symmetry point groups we will use matrices to represent 02:03.540 --> 02:09.619 the symmetry operations and thereby we will be representing the whole group by those 02:09.619 --> 02:12.480 set of matrices that actually represent the symmetry operation and then we will 02:12.480 --> 02:20.470 we will use some other properties of those matrix representation to make our life more 02:20.470 --> 02:32.990 simple and the so lets lets try to learn about this representation of a group and 02:32.990 --> 02:42.129 before we go and the form the matrices for symmetry operation and there by the whole 02:42.129 --> 02:48.970 representation of a group in transfer matrix we will try to learn little bit thing 02:48.970 --> 02:55.629 about matrices matrix algebra many of you already know matrix algebra but for those 02:55.629 --> 03:04.290 who doesnt know matrix algebra or have learnt sometime back at once to you know 03:04.290 --> 03:13.219 have a you know then for them we will go through the basic matrix algebra here so we will 03:13.219 --> 03:19.599 give a briefly review of matrix algebra in this section so first we will deal 03:19.599 --> 03:30.770 with certain definitions so first case we have to know what is a matrix so if you look 03:30.770 --> 03:33.599 at your screen definition of matrix its given there which says but a matrix is a collection 03:33.599 --> 03:40.189 of number ordered by rows and columns ok so if you have some numbers some digits which 03:40.189 --> 03:48.069 are order in rows and some columns so its array and how it is presented its presented 03:48.069 --> 03:51.870 in terms of you know its like the numbers in rows and columns they are written within 03:51.870 --> 03:55.670 parenthesis ok so that what is also mentioned here that the element of the matrix 03:55.670 --> 04:02.859 so all the numbers that are there in any other rows or columns they called element of matrix 04:02.859 --> 04:08.249 so all this element will be in parentheses bracket or braces so an example is is shown 04:08.249 --> 04:15.409 here where this x which is written in bold term so normally when you know use a 04:15.409 --> 04:21.300 symbol for any matrix then you use either a bold or you know bold italics or some you 04:21.300 --> 04:28.960 know particular type of scrip ok so see this x is a matrix which is you know within 04:28.960 --> 04:34.860 in this way so you can see there are two columns one here ones here correct and there 04:34.860 --> 04:42.949 are three rows sorry this two are rows i am sorry and the here are the columns so 04:42.949 --> 04:48.360 i have three columns so this matrix x has two rows and three columns 04:48.360 --> 04:53.080 now this this is general example of matrix now if by any chance a matrix has you 04:53.080 --> 04:57.240 knows n number of rows and n number of columns that is the number of rows and numbers 04:57.240 --> 05:03.960 of columns are equal then we call that matrix as a square matrix ok now an example of square 05:03.960 --> 05:10.099 matrix is given here a is a square matrix because it has two columns and two rows so 05:10.099 --> 05:16.870 number of columns rows are equal so this a is an is a square matrix while if i look 05:16.870 --> 05:21.889 at the matrix b here it has two columns but three rows similar to the example shown in 05:21.889 --> 05:31.599 the top so this b or this x here they are not square matrices ok 05:31.599 --> 05:39.460 now one more types of matrix that we can think about is a symmetric matrix so a 05:39.460 --> 05:55.870 symmetric matrix is a square matrix in which x i j equals to x j i so this i and j are 05:55.870 --> 06:05.470 used for noting the rows and columns so ith and jth column is given by x i j and 06:05.470 --> 06:13.490 the jth and ith columns is given by x j i ok and x is any element ok so as whole any 06:13.490 --> 06:21.030 element is any element of a matrix is given by if x matrix element is x in its x i 06:21.030 --> 06:29.169 j but i is rows and j is a columns so for all i and j if this equality holds that 06:29.169 --> 06:36.080 x i j equal to x j i then the matrix is called as square matrix ok 06:36.080 --> 06:42.510 so if you look at here the element saying one one ok corresponding to row one column 06:42.510 --> 06:44.669 one is nine and the you know the element corresponding to row one column two is one 06:44.669 --> 06:49.860 on the other hand if you look at the matrix b you will see that the element for which 06:49.860 --> 06:56.180 the row two column one it is just opposite of the previous example you know is the same 06:56.180 --> 07:02.810 so this a is an example of the symmetric matrix while if you look at the b then you 07:02.810 --> 07:09.909 will say this you know this is not equals to this ok so that means here i have one two 07:09.909 --> 07:23.159 this is a element one two is not same has two one ok but here this one two is equal 07:23.159 --> 07:33.319 to two one that is one fine some more definitions here so what is an identity matrix identity 07:33.319 --> 07:39.030 matrix is that particular matrix for which the all the diagonal elements are unity ok 07:39.030 --> 07:43.349 so you know rows and columns so if you go to the along the diagonal if will be like 07:43.349 --> 07:50.469 you know element one one and two two three three and so on ok so this for square matrix 07:50.469 --> 07:56.469 this is the diagonal correct this just like a take a square and you get a diagonal for 07:56.469 --> 08:04.719 that so along the diagonal if you look at you will see that all the elements are unity 08:04.719 --> 08:12.289 and rest of the elements are zero so this is called identity matrix and all the time 08:12.289 --> 08:16.660 identity matrix is denoted by this symbol i capital i ok this is a normal convention 08:16.660 --> 08:21.169 ok so now if you have two matrices and 08:21.169 --> 08:24.879 you want to do subtraction or you know addition with those two matrices how do you 08:24.879 --> 08:32.710 do that so to add two matrices then you know you cannot do it with with any two matrix 08:32.710 --> 08:38.250 ok so they both must have the same number of rows and they both must have the same number 08:38.250 --> 08:45.540 of columns so if i have a say two by three matrix and then i have a four by four matrix 08:45.540 --> 08:52.060 then i cannot do any kind of subtraction or addition using this two matrices ok the reason 08:52.060 --> 09:01.050 will be you know like this because for any addition the elements of two matrices 09:01.050 --> 09:08.339 are added together and this is done by element by element ok so if i have matrix like here 09:08.339 --> 09:14.110 from here so i have like in the first row one line and minus two and on on the other 09:14.110 --> 09:20.990 hand other matrix i have eight minus four three so i can map one two one so one plus 09:20.990 --> 09:28.279 eight nine plus minus four and minus two plus three so i can have this one two one addition 09:28.279 --> 09:34.260 or one two one subtraction but if would have matrix something like you know one two 09:34.260 --> 09:41.420 three four five six seven eight nine i could not add this two this matrix because the first 09:41.420 --> 09:52.170 row second row would be fine but the third row it will not work so i cannot this element 09:52.170 --> 10:00.320 by element addition so that is the condition of this you know addition and subtraction 10:00.320 --> 10:05.920 and how you do it i told you and this is specifically mentioned here ok and this is 10:05.920 --> 10:09.460 true for both subtraction as well as addition ok 10:09.460 --> 10:12.060 now whatever matrix multiplication because this is something which we will use very 10:12.060 --> 10:14.820 often in case of good theory when we use this matrices to represents symmetry operations 10:14.820 --> 10:19.399 we need to do you know like two success operations so it means like you know operation 10:19.399 --> 10:23.740 a followed by operation b so when we will try to use matrices to represent this symmetry 10:23.740 --> 10:26.310 operations then this matrices will be multiply ok and thats why this quite important 10:26.310 --> 10:28.681 in outr context so there are certain rules for matrix multiplication which are you 10:28.681 --> 10:34.110 know scripted here so the first one it consist the multiplications between the matrix 10:34.110 --> 10:41.139 and the scalar so if you are given any matrix or any order and you have given the 10:41.139 --> 10:47.100 scalar and you have to multiply the scalar to this in a matrix then its simply multiplying 10:47.100 --> 10:58.459 each element of the matrix by this scalar quantity then you get a multiplication ok 10:58.459 --> 11:09.330 so for you know any matrix b whose element is b i j and if you have given given 11:09.330 --> 11:15.170 as scalar quantity a then the new matrix that we form after the multiplication its element 11:15.170 --> 11:23.290 r i j will be related to a and b i j and as follows ok so its essentially multiply all 11:23.290 --> 11:29.770 the element of the matrix b by test scalar quantity so one example is given here were 11:29.770 --> 11:34.260 you can see that eight is a square is a scalar quantity and you have this two by two matrix 11:34.260 --> 11:41.390 so eight is been multiplied with each of the element so eight two you get sixteen eight 11:41.390 --> 11:46.440 six you get forty eight and so on ok so one important aspect of this scalar multiplication 11:46.440 --> 11:50.769 with matrix is it its commutative nature so it doesnt matter whether you multiply 11:50.769 --> 12:00.339 the you know matrix with this scalar or you know scalar with matrix so order doesnt matter 12:00.339 --> 12:06.430 whatever we do the result will be the same so that is stated here a b equals to b 12:06.430 --> 12:14.019 a now the most important rule here these 12:14.019 --> 12:18.980 as follows so it in this rules involves the multiplication of a row vector by a column 12:18.980 --> 12:25.410 vector ok now to perform this the row vector must have has many columns has the columns 12:25.410 --> 12:32.130 vector has ok so it says that if you take any row vector ok so its just a row ok that 12:32.130 --> 12:39.320 is called a row vector so you have like you know one row and n number of columns so 12:39.320 --> 12:43.529 you dont have more than one rows so this is called a row vector and the column vector 12:43.529 --> 12:52.320 is just as you know one column and you can have n number of rows so only when you can 12:52.320 --> 12:56.470 have this multiplications when this row vector has the number of columns which is equal to 12:56.470 --> 13:03.730 the number of rows that the column vector has ok so that is shown by an example here 13:03.730 --> 13:10.490 so here this is a row vector that has three columns it can be multiplied with a column 13:10.490 --> 13:17.550 vector that has three rows if i would have two rows from the column vector or four rows 13:17.550 --> 13:24.040 from the column vector i could not do this multiplication ok so this is very very important 13:24.040 --> 13:27.190 rule and this immediately tells you that you cannot have multiplication between any two 13:27.190 --> 13:35.769 matrices that has to follow this criteria so this has been shown by a by column vector 13:35.769 --> 13:40.399 and row vector but this applicable to the matrix as a whole so all the other type 13:40.399 --> 13:46.170 of matrix multiplication involve the multiplication of a row vector and a column vector so you 13:46.170 --> 13:50.570 can you can you know imaging when you given matrix of whatever order as you know as 13:50.570 --> 13:56.459 a you know on symbol of several row vectors several column vectors ok so in general 13:56.459 --> 14:05.089 if you want to do matrix multiplication so it will just involve multiplication of a row 14:05.089 --> 14:10.720 vector and the column vector like we showed just in the previous slide 14:10.720 --> 14:19.730 so specifically in the expression r equal to a b where a and b are two matrices 14:19.730 --> 14:27.170 and r is the resulted the matrix of the multiplications and the you know this this small r and 14:27.170 --> 14:34.140 and small b they represents the elements of those matrices r a and b which are been in 14:34.140 --> 14:41.950 capital and in bold form so here you can see that this r i j equal to a i dot and 14:41.950 --> 14:52.130 b dot j where this a i dot is the ith row vector in matrix a and b dot j is the jth 14:52.130 --> 14:58.690 column vector in matrix b now you see that you know the number corresponding to 14:58.690 --> 15:07.560 the to the column for a matrix is give as dot and same dot is given for the you know 15:07.560 --> 15:11.710 the number of row for the element corresponding to b matrix so this two dots are same meaning 15:11.710 --> 15:16.009 that their numbers are same and then only we can do the multiplication so following 15:16.009 --> 15:24.100 that you have an example here where you have taken this a and b and a is matrix whose 15:24.100 --> 15:28.580 as two row three columns while b is a matrix having three rows and two column so you 15:28.580 --> 15:35.089 can do a multiplications like this so rule of when you do multiplication you know 15:35.089 --> 15:38.280 multiply one matrix to another you do it in this way 15:38.280 --> 15:45.540 so first you you know follow this and you follow this ok next again you you know multiply 15:45.540 --> 15:52.470 with this one then you do with using this one and then again this one ok so we follow 15:52.470 --> 15:58.980 like a row and then column ok and you have it like here like you know one element 15:58.980 --> 16:02.920 multiplied to another element and then you keep doing the multiplication and some then 16:02.920 --> 16:09.170 all together which is explicitly shown here so this is the you know resulting element 16:09.170 --> 16:14.160 r one one and this how you do the multiplication so one two one corresponds here and ultimately 16:14.160 --> 16:18.790 you get the respective values which will active has the element of the resultant matrix 16:18.790 --> 16:26.110 so you get r one one and r one two so one one as i said it follows the first row of 16:26.110 --> 16:29.980 the first matrix on multiplies with the first column of the second matrix and r one two 16:29.980 --> 16:36.930 will be the first row of the first matrix multiplied to the second column of the second 16:36.930 --> 16:43.200 matrix ok in that way it will keep going and ultimately you will get the all the elements 16:43.200 --> 16:47.829 of the resulted matrix and ultimately you have product like this 16:47.829 --> 16:54.110 now this is interesting right you had like matrix with three two rows and three 16:54.110 --> 16:58.730 columns which multiplies with matrix having three rows and two columns and then the 16:58.730 --> 17:04.890 produce is two by two matrix ok having two row and two columns so for matrix multiplication 17:04.890 --> 17:15.400 to be legal the first matrix must have has many columns has the second matrix has rows 17:15.400 --> 17:26.130 that your probably understood by now we will get the example of the you know condition 17:26.130 --> 17:34.799 of the row vector and column vector if you just extent that to any given form of 17:34.799 --> 17:40.070 matrix then you come to this particular rule which was talking about right now so for again 17:40.070 --> 17:45.390 if we state the sentence so for the matrix multiplication to be legal the first 17:45.390 --> 17:52.360 matrix must have has many columns as the second matrix has rows this of course is the requirement 17:52.360 --> 17:57.010 for multiplying a row vector by a column vector right 17:57.010 --> 18:04.460 the resulting matrix will have has many rows has the first matrix and has many columns 18:04.460 --> 18:07.690 has the second matrix so thats why you are getting this two by two matrix because the 18:07.690 --> 18:11.630 first matrix had rows and second matrix had two column so number of rows of the first 18:11.630 --> 18:15.110 matrix and number of column of the second matrix will give you the total dimension of 18:15.110 --> 18:20.940 the matrix that is resultant ok so thats about the matrix multiplication 18:20.940 --> 18:27.130 now there are certain other properties of matrix and that we will look at now 18:27.130 --> 18:32.770 one is the transpose of the matrix ok this is the name something which is important in 18:32.770 --> 18:41.830 our particularly particularly our context so the transpose matrix is a denoted by 18:41.830 --> 18:50.950 you know it superscript t in some time it is also a prim ok or with a superscript 18:50.950 --> 19:03.260 small t but normally we use a with a superscript t plus t is capital letter alright so thats 19:03.260 --> 19:11.550 the symbol of the transpose and now what is the transpose so the first row of a matrix 19:11.550 --> 19:20.700 becomes a first column of the ya i know transpose matrix so i had say you know matrix having 19:20.700 --> 19:24.590 three rows and three columns ok or say three row and two column ok so this matrix 19:24.590 --> 19:27.940 when i will transpose it will become like you know matrix having two rows and 19:27.940 --> 19:32.810 three columns so here is one example where a is a matrix 19:32.810 --> 19:39.610 having two rows and three columns and that when i transpose it it becomes three rows 19:39.610 --> 19:46.530 and two columns ok so the transpose of a row vector will be a column vector obviously and 19:46.530 --> 19:52.310 the transpose of a column vector will be away vector as to be the transpose of a symmetric 19:52.310 --> 19:59.580 matrix is simply the original matrix so you know in symmetric matrix what we found 19:59.580 --> 20:08.310 that any r i j equals to r j i ok if r i j and r j i are the element of the matrix so 20:08.310 --> 20:14.950 in that case if you know if just make a transpose you are not going to get something 20:14.950 --> 20:19.809 new so for symmetric matrix the transpose will return you the same matrix alright 20:19.809 --> 20:27.240 next we need to know what is the inverse of the matrix so in scalar algebra the 20:27.240 --> 20:30.530 inverse of a number is that number which when multiplied by the original number gives a 20:30.530 --> 20:37.800 product of the unity hence the inverse of x is simply one upon x because on the multiplication 20:37.800 --> 20:46.400 x into one upon x will give me one or in slightly different notation i can write it x inverse 20:46.400 --> 20:51.980 ok so x inverse we are familiar with that already now in matrix algebra the inverse 20:51.980 --> 21:01.350 of the matrix is that matrix which when multiplied by the original matrix gives an identity so 21:01.350 --> 21:10.059 the notation remains a same so here in case of matrix algebra what you have have a 21:10.059 --> 21:16.210 matrix and if i need to find out the inverse matrix that the inverse matrix will be such 21:16.210 --> 21:22.360 that if i multiply this original matrix with this inverse i will get an matrix which is 21:22.360 --> 21:29.020 identity matrix ok so the inverse of a matrix is denoted by the superscript inverse so its 21:29.020 --> 21:38.760 a inverse sine so if i write same matrix by capital l a in bold letter so inverse of 21:38.760 --> 21:44.370 the matrix will be capital a in bold letter inverse ok 21:44.370 --> 21:54.610 so its implied that the you know a a inverse is equal to a inverse a that is they commute 21:54.610 --> 22:00.029 and the result is identity and the in order to have and inverse rather wise your for the 22:00.029 --> 22:05.330 inverse of a matrix exist the matrix must be a square matrix that means the number of 22:05.330 --> 22:12.580 rows and numbers of columns must be equals but the reverse not always true so all 22:12.580 --> 22:19.770 the square matrices may not have inverse ok in some case inverse does not exist 22:19.770 --> 22:24.710 now before we go to the representation of group using matrix we learned couple of 22:24.710 --> 22:38.570 more things one is the trace of the matrix this is something which you will be using 22:38.570 --> 22:43.170 throughout the course hence for ok once you go to there it representation so trace of 22:43.170 --> 22:54.940 a matrix is is the sum of the diagonal elements ok of the matrix so if i look at 22:54.940 --> 23:03.809 any given matrix say on the matrix on the screen here the sum of three six and minus 23:03.809 --> 23:17.820 five will give the trace of the matrix ok so this has you know very important consequence 23:17.820 --> 23:23.929 particularly under context of you know symmetric point groups and then representation 23:23.929 --> 23:37.380 and you will soon learnt that this trace of matrix particularly certain matrix which 23:37.380 --> 23:50.110 can be in a block factor to a form by you have only the diagonal elements and the off 23:50.110 --> 24:04.580 diagonal elements are zero in this cases this trace of those matrices will be some word 24:04.580 --> 24:09.820 characteristics of this particular matrices that in term represent certain symmetry 24:09.820 --> 24:18.690 you know operation or you know in case of group we are talking about then it will represent 24:18.690 --> 24:27.950 the you know any particular group element so this case in the context of group theory 24:27.950 --> 24:35.179 we will be using the term character ok but till now we will just you know stick 24:35.179 --> 24:45.899 to this terminology face until unless we go to the representation of the group and we 24:45.899 --> 24:49.610 will also learn then why these traces call the character ok but for your information 24:49.610 --> 24:55.340 you should know the this trace will be actually acting as character of any representation 24:55.340 --> 24:59.850 very well and the one more thing i would like to 24:59.850 --> 25:45.841 mention here is about the determinant of a matrix so how many of already know what determinant 25:45.841 --> 26:20.120 is so the determinant is a useful value that can be computed from the elements of a square 26:20.120 --> 26:35.200 matrix 26:35.200 --> 27:03.070 so the determinant of a matrix a is denoted as either this term or this symbol ok in 27:03.070 --> 27:25.120 the case of a two by two matrix the specific formula for the determinant is simply the 27:25.120 --> 27:39.370 upper left elements times the lower right element minus the product of the other two 27:39.370 --> 27:55.150 elements ok so if you have a three by three matrix then we can determine the value 27:55.150 --> 28:27.850 of this whole determinant in this fraction ok so you have this a fix and then you find 28:27.850 --> 28:41.930 out about the this particular part ok and then you subtract you know part where you 28:41.930 --> 29:06.789 know b is multiplied with the determinant which is consist of this two terms ok on 29:06.789 --> 29:25.500 your screen here you can seen and then you do it for the c term ok so in this 29:25.500 --> 29:46.669 way you can find out about the determinant of any order so if you have a square matrix 29:46.669 --> 30:02.140 of any given order we can find the value for that you know 30:02.140 --> 30:11.350 determinant ok so with that we will stop here today 30:11.350 --> 30:16.990 and in the next class we will come back with same representation of the symmetry operations 30:16.990 --> 30:23.990 where you will use this you know this brief knowledge of matrix algebra thats we 30:23.990 --> 30:34.440 learnt today and we used it and try to find out about you know representation of the 30:34.440 --> 31:02.429 symmetry operations thank you for your attention see you tomorrow