OK, this is our lecture where complex numbers come in.
It’s …complex number had slipped in this course because even a real Matrix can have complex eigenvalues, so we meet complex numbers there, as the eigenvalues, and complex eigenvectors.
And we…a…this is properly the last…we have a lot of other thing to do about eigenvalues and eigenvectors, and that will be mostly real. But at one point somewhere, we have to see what would you do when the numbers become complex numbers. What happens, when the vectors are complex, when the matrices are complex, when the…what’s the inner product of two, the dot product of two complex vectors. We just have to make the change, just see what is the change when numbers become complex.
Then, can I tell you about the most important example of complex matrices? It comes in the Fourier matrix. So the Fourier matrix which on described is the complex matrix it’s certainly be most important complex matrix, is the matrix that we need in the Fourier Transform, and the really, the special thing that I want to tell you about is what’s called The Fast Fourier Transform, and every body refers to the FFT, and it’s in all computers, and it’s used.. is been used as we speak in a thousand places, cause it likes transformed whole industry, to be able to do the Fourier transform fast,
Which means multiplying, how do I multiplying fast by that matrix, by that… by that n by n matrix, normally multiplications by a n by n matrix would normally be n squared multiplications. Cause I’ve got n squared entry and none of them are zero. This is a full matrix, and it’s a matrix with orthogonal columns. I mean it’s just like the best matrix. And this Fast Fourier Transform idea reduced this it’s n squared which was slowing up, the calculation of Fourier transform down to n*log (N), n log n, log to the base of 2 actually, and is this, when that hit, when that possibility hit. It made a big difference, everybody …realized, gratulated what …a… That is a simple idea, you see it’s just a simple matrix factorization… but it changed everything.
Ok, so I want to talk about complex vectors, and matrices in general, we cap a little bit from last time, and … the Fourier matrix in particular.
Ok, so what’s the deal, all right, the main point is what about length. I…I’m given a vector, I have a vector x, or let me call that z, as that remained us it’s complex for the moment. But I can … later I will call that component x, x will be complex numbers. But it’s a vector z1, z2…down to z (n). So the only novelty is it’s not in R (n) anymore. It’s in complex n dimensional space. Each of those numbers is a complex number. So this z, z1 is in C(n), n dimensional complex space instead of R (n). It’s just a different letter there. But now the point about its length is…is what. The point about its length is that ZT z is no good. ZTZ, if I just put down ZT here. It would be z1, z2, …to Z (n). Doing that multiplication doesn’t give me the right thing. Why not? Because the length squared should be positive.
And If I multiply… suppose this is like 1 and i. What the length of the vector with components 1 and i? What if I do this? So in it just 2, I mean C (2), two-dimensional space, complex space. With the vector whose components are 1 and i. All right, so if I took one times one, and i times i. And added, ZTZ would be … zero! But I do know that vector (is not) doesn’t have length zero, the vector have components 1 and i.
This multiplication, what I really want is z1 conjugate z1. Do you remember what z1 conjugate z1 is? So you see that first step will be z1 conjugate z1, which is the magnitude of z1 squared, which is what I want. That’s liked three squared or five squared. Now if it’s … if z1 is i, then i multiplied by minus i gives 1.plus 1. So the component of length, the component i, its macular squared is plus one, that’s great.