Now that you know how to load and save data and put your data in matrices and so on, in this video I'd like to show you how to do computational operations on data. And later on we'll, we'll be using the source of computational operations to implement learning algorithms. Let's get started. Here's my Octave window. Let me just click the initialize some variables, to use for our examples. Set A to be a two by two matrix, and set B to a two by two matrix, and let's set C to a 2 by 2 matrix, like so. Now, let's say I want to multiply two of my matrices. So, let's saw I want to compute a * c I just type a * c. So it's a three by two matrix * a 2 by 2 matrix. This gives me this 3 by 2 matrix. You can also do element wise operations. And do a. * b. And what this will do is, it'll take each element of a, and multiply it by the corresponding elements of b. So that's a * b. That's a. * b. So, for example, the first first element gives. One times eleven which gives eleven. The second element gives 2 * 12 which gives 24 and so on. So does the element wise multiplication of two matrices and in general the p grid tends to, is usually used to denote element wise operations and auto. So here's a matrix A and Im going to do A.iii this gives me the mul- the element wise squaring of A so you know, one square is one, two square is four and so on. Let's set V to a vector, let's set V as 1, 2, 3 as a colon vector. You can also do one dot over V to do the element Y's reciprocal of V. So this gives me 1 / 1, 1 / 2, and 1 / 3. This works super matrices, so 1. / over a gives me that element y, your inverse, of a. and once again the p-ratio gives us a clue that this is elements wise operation. You can also do things like log V, this is element wise logorithm of the of the, the, V, E to the V, let's see of, base E exponentiation of v elements. So, this is E, this is E squared, E cubed, because, this was V, so this, and, I can also do absent V, to take the element wise absolute value of V, so here, you know V was all positive to aps take minus one to minus three. The element wise absolute value gives me back these non-negative values and negative v gives me the minus of v. This is the same as negative one times v, but usually you just write negative v instead of negative one times v. And . What else can you do here is another neat trick. So lets see, lets say I want to take V and increment each others elements by one, well what we need to do is by constructing a three x one. Vector this old ones and adding that to V, so if I do that this increments v by from 1, 2, 3 to 2, 3, 4. The way I did that was length of b is 3 so 1's length would be by one. This is let 1's of 3 by 1 So, that's 1's three by one. On the right, and what I did was V plus one is V by one, which is adding this vector of all 1s to V and so this increments V by one and you, another simpler way to do that, is actually type V plus one, right, so just V and V plus one also means to add to one element wise to each of my elements of V. Now. Let's talk about more operations. So here's my matrix A. If you want to write A transpose the way to do that is to write A prime. That's the, apostrophe symbol. It's the left quote. So, in, in on your keyboard you probably have a left quote, and a right quote. So this is, excuse. This is actually a standard quotation mark. Is, type A transpose this gives me the, you know, the transpose of my matrix A. And of course, A transpose. If I transpose that again then I should get back my matrix A. Some more useful functions let's say lowercase a us 115 20.5, so it's a one by four matrix. Let's say I say vow is max of A, this returns the maximum value of A which in this case is fifteen and I can do vow end max A and this returns vow and end which are going to be the maximum value of A which is fifteen as well as the index. So is the element number two of A that was fifteen so ends is my index in this. just as a warning that you do max A. Where A is a matrix, what this does is this, this actually does the column wise maximum, but I'll say a little bit more about this in a second. So using this example, the variable lower case, a. If I do a less than three this does the element y's operation, element y's comparison. So the first element of a is less than three so this gives us one, second element of a is not less than three so this device is zero cause it's false. The third and fourth elements of r I meant less than three, 34 elements are less than three so there's just one, one, so this does the elementwise comparison of all four elements of the variable lowercase A to three and the returns true or false depending on whether or not it's less than three. Now, if I defined a less than three, this will tell me which are the elements of a. The, the variable A are less than three, and in this case the first, third and fourth elements are less than three. For my next example, let me set A to be equal to magic to be the magic function, returns, let's say, help magic. The magic function returns functions called magics returns these matrices call magic squares, they have this, yeah mathematical property that's all of their rows and columns and diagonals sum up to the same thing. So you know, it's not actually useful for machine learning as far as I know but I'm, I'm just using this as a convenient way. You know, to generate a 3 by 3 matrix. And, and these magic squares anyway we have the property that each row, each column and the diagonals all add up to the same thing. So it's kind of a, mathematical construct. I use magic, I use this magic function only when I'm doing demos or when I'm teaching Octave, like this and I don't actually use it for any, you know useful machine only application. But let's see if I type RC equals find A greater than or equals seven, this finds all the elements of A, that are greater than or equal to seven. And so, RC sends row and columns. So the 1-1 element is greater than seven. The 3-2 element is greater than seven and the 2-3 element is greater than seven. So let's see. The 2-3 element, for example, is a 2-3 is seven is this element. Out here, and that is, indeed, greater than or equal to seven. By the way, I actually don't even memorize, myself, what these fine functions do. And what all of these things do myself. And whenever I use the fine function, sometimes I forget myself exactly what it does. And you know, I to, like, help find, to look up the document. Okay just two more things and I'll quickly show you, one is the sum function. So here's my A I'm going to type sum A which adds up all the elements of A and if I want to multiply them together I type prod, A prod sense of product and this returns the product of these four elements of A. four A rounds down these elements of A so 0.5 is rounded down to zero and seal or sealing A gets rounded up to the nearest integers so 0.5 gets rounded up to 1. You can also let's see. Let me type rand(3), this generates a 3 by 3 matrix. If I type max rand(3), rand(3) what this does is it takes the element wise maximum of a two random 3 by 3 matrices. So, you notice, all of these numbers tend to be a bit on the large side. Because each of these is actually the max of a randomly of, of element Y's max of two randomly generated matrices. This is, this was my magic number, this was my magic square three by three A. Let's say I type max A and then this will be eight. Open close square brackets, comma one. What this does is this takes the column-wise maximum. So, the max of the first column is eight, the max of the second column is nine, and the max in the third column is seven. This one means to take the max along the first dimension of A. In contrast if I were to take max A, this funny notation too, then this takes this per row maximum, so the max of the first row is eight, max of second row is seven, max of the third row is nine and so this allows you to take maxes, either, you know, per row, or per column. And if you want to and remember, it defaults to a column like wise element. So you won't find. If you want to find the maximum element in the entire matrix A. You can type max of max of A. Like so, which is 9 or you can turn A into vector and type max of A: like so. And this treats this as a vector. And takes the max of, max elements of that vector. Finally, let's set A to be a 9 by 9 magic square, so remember the magic square has this property that every column and every row sums the same thing and also the diagonals. So here's a nine by nine may magic square, so let me do sum A1 so this does a per column sum so I'm going to take each column of A and ad them up and this, you know let's just verify that indeed for a nine by nine magic square every column adds up to 369, adds up to the same thing. Now let's do the row-wide sum so the sum A, comma 2 and this sums up each row of A, and indeed each row of A also sums up to 369. Now, let's sum the diagonal elements of a and make sure that they, that, that also sums to the same thing. So, what I'm going to do is construct a 9 by 9 identity matrix. That's I9 and I'm going to take a and construct, you know multiply A, element-wise. So here's my matrix A. I'm going to do a * I9. And what this will do is take the element-wise product of these two matrices. And so this should wipe out everything in a except for the diagonal entries. And now I'm going to do sum, sum of a of that. And this gives me the sum of this, these diagonal elements. And indeed, it is 369 you can sum up the other diagonal as well. So the sort of top left to bottom right, you can sum up the opposite diagonal from bottom left to top right. the sum, the, the commands for this is somewhat more cryptic. You don't need to know this. I'm just showing you this in case any of you are curious. But let's sese flip U, flip UD stands for flip up down. But if you do that, that turns out to sum up the elements and the opposites of the other diagonal that also sums up to 369. Here let me show you. Whereas I9 is this matrix. Flip up down of I9 you know takes the identity matrix and flips it vertically. So you end up with, excuse me flip UD end up with 1's on this opposite diagonal as well. Just one last command and then that's it, and then that'll be it for this video. Let's set A to be the magic, three by three magic square game. If you want to invert a matrix, you type P inv A. this is typically called the pseudo inverse, but it doesn't matter. Just think of it as basically the inverse of A, and that's the inverse of A. And so I can set, you know, temp equals P inv of A and [INAUDIBLE] temp times A. This is indeed the identity matrix with essentially ones on the diagonals and zero. It was on the off diagonals up to a numerical round off. So that's it for how to do different computational operations on the data matrices. and after running a learning algorithm, often one of the most useful things is to be able to look at your results. Or to plot or visualize your results. And in the next video I'm going to very quickly show you how again with one or two lines of code, using Octave. You can quickly visualize your data or plot your data. And use that to better understand you know, what your learning algorithms are doing.
