Now, it's going to be really useful if we can make a transformation matrix whose column vectors make up a new basis, all of whoms't vectors are perpendicular or what's called Orthogonal to each other. In this video, we're going to look at this and why this is. But first, I want you to find a new operation on the matrix called the transpose. This is where we interchange all the elements of the rows and columns of the matrix, so I'll denote the transpose as T and I'll say that the ijth element of the transpose of A, is equal to the elements on the opposite diagonal A_ji. So if I had a matrix like 1,2,3,4, and I wanted to know its transpose, then I will interchange the elements that are off the diagonal. So the one and the four stay where they are. Because if is, I find i and j are the same; 1,1 1,1 they would stay the same, the same for 2,2. But the element 1,2 interchange to the element 2,1 so they go like that. So 1,2,3,4 becomes, when I transpose it 1,3,2,4. Now, let's imagine that I have a square matrix of dimension n by n, it defines a transformation with a new basis and the columns of that transformation matrix A. Some vectors a_1, a_2 that are the basis vectors in the new space as we've said before and I'm going to make this a special matrix while I impose the condition that these vectors are orthogonal to each other and they have unit length. That is, a_i dotted with a_j is equal to zero if i isn't equal to j, that is they're orthogonal, and it's equal to one if i is equal to j, that is they are of unit length. Now, let's think about what happens if I multiply this matrix on the left by a transpose. So A transpose is going to be given by just flipping the rows and columns of all of the elements across the leading diagonal of A. So that is, I'm going to have a_1 is going to become a row, a_2 is going to become the next row all the way down to a_n because I flip all of the elements across the leading diagonal. So while I multiply A_T by A, let's see what happens, it's going to be quite magical. So if I get A_T times A, then I'm going to get, well, the first element here is going to be the first row times the first column, and that's a_1 dotted with a_1, and that's one. The second element here, the first row and the second column is a_1 dotted with a_2 and that's zero, and a_1 dot a_3 all the way up to a_n is going to give me a whole series of zeros where this guy remember is a_n. Then when I do the second row on the first column, I'm going to get another zero. I want to do the second one and the second column I'm going to get a one. Second row, third column zero all the way across. I want to do the third one, I'm going to get the same again, I'm going to get zero zero. It only says it's a_3 times a_3, and that's going to be that element all the way across and what I'm gradually building up here, is I'm building up the identity matrix. So what I found is that A_t times A gives me the identity matrix, and what that means is, is that A_T is a valid inverse of A. Which is really kind of neat. A set of unit length basis vectors that are all perpendicular to each other are called an orthonormal basis set, and the matrix composed of them is called an orthogonal matrix. One thing also to know about an orthogonal matrix is that because all the basis vectors, any of unit length, it must scale space by a factor of one. So the determinant of an orthogonal matrix must be either plus or minus one. The minus is what arises in the new basis, if the new basis vector set flits space around. They stay inverted, they make it left-handed. Notice that if A_T is the inverse then I should be able to post multiply A by A_T and get the identity. So I can do this on the way round. Which also means by the same logic that the rows of the orthogonal matrix are orthonormal, as well as the columns which is neat, and we saw in the last video that actually the inverse is the matrix that does the reverse transformation. So the transpose matrix of an orthonormal basis vector set is itself another orthogonal basis vector set which is really neat. Now, remember that in the last module on vectors we said that transforming a vector to a new coordinate system was just taking the projection or dot product of that vector, so say that's the vector, with each of the basis vectors so basis factor this one, basis factor that one, and so on so long as those basis vectors were orthogonal to each other. If you want to pause and think about that for a moment in the light of all we've learned about matrices, just look and think about this for a moment. Now in data science what we're really saying here is that wherever possible, we want to use an orthonormal basis vector set when we transform our data. That is, we want our transformation matrix to be an orthogonal matrix. That means the inverse is easy to compute. It means the transmission is reversible because it doesn't collapse space. It means that the projection is just the dot product. Lots of things are nice and pleasant, and easy. If I arrange the basis vectors in right order, then the determinant is one, and that's an easy way to check and if they aren't just exchange a pair of them and actually then they will be determinant one rather than the minus one. So what we've done in this video is look at the transpose and that's led us to find out about the most convenient basis factor set of all which is the orthonormal basis factor set which together make the orthogonal matrix whose inverse is its transpose. So that's really cool.
