 The one of the key consequences of the spectral theorem is that it allows us to define different types of inner products for any humeration matrix so if a satisfies a Equals a star a is our mission matrix then define Let's call the inner product of V and W sub a as V star a W Note that our traditional inner product VW Is really VW I and certainly the identity is an information matrix So why is it that I should call this an inner product? Why is this a reasonable operation to do? Well, the thing is that it satisfies the key properties that we want an inner product to have First of all, it's certainly multi-linear So it's based on matrix multiplication. If I say scale V or W the result scales appropriately if I say V1 plus V2 and Insert that here or W1 plus W2 I can distribute it as appropriate and It satisfies this key property that V W a Is W VA up to conjugation Here's a quick proof of that V star a W is a complex number and for a complex number Taking a conjugate is the same thing as taking a star If I think as a one-by-one matrix That's W star a star V star star and that of course is W star a star is equal to a and V star star is V and We're done To game to gain some intuition about this inner product V. W a This is where the spectral theorem comes in so because a can be written as a Real diagonal matrix altered by a unitary change of basis That means we can write the following At first this looks just like some algebraic manipulations When we come to the end, you'll see that's really a very geometric statement in other words if I want to take The bracket VW with respect to a I can compute this way in some orthogonal basis you I essentially taking the standard Hermitian product of V and W in that new basis But I'm stretching the different directions by different factors lambda one lambda two up to lambda n in other words If we choose all the lambdas to be one and have you be the standard orthogonal basis Then this is the standard inner product However, if I choose some other lambdas and some other orthogonal basis I get another notion of an inner product The only condition you might want to have is that you might you might insist that Lambda one up to lambda n are all positive Remember the spectral theorem guarantees that they are already real So we don't have to worry about having a manager answers here But you might want to insist that they're all positive in this in that case we call this positive definite Our Hermitian matrix is called positive definite If all of its eigenvalues are strictly positive the word positive definite is a little strange that's sort of a historical artifact if for instance of all the eigenvalues were a Greater than or equal to zero. It's sometimes called positive indefinite. So it's like these are definitely positive or they are Indefinite would mean non-negative, right? So it's kind of a strange terminology, but that's the top terminology that stuck So the point is that what this format allows us to do with the spectral theorem is say Choose any orthogonal basis you like and choose any sort of scaling factors in those directions and you get a new inner product Corresponding with that choice and Each such choice is really saying user Hermitian matrix. I made a glib comment in the previous video about how general relativity depends on The spectral theorem and I want to take out just a moment to explain what I mean by that The key idea behind Einstein's theory of general relativity is that we have two ideas that mass and energy Bend space Or actually bend spacetime I'll say space for now and space Tells objects how to move So the second one is really what I want to focus on what do we mean when you say space tells objects how to move? well at any given location You can think it was a orthogonal frame of reference and say say how far away are two points near here And you use the notion of a norm So our notion of measurement of how far away things are and how long it takes to get somewhere and how fast we're going All those ideas come from the inner product at a point now The thing we know about the universe is that it's not the same at every point We have a notion of gravity is stronger in some places versus other places and we want to express that as a geometric idea so if we sort of take a picture of a Section of the universe with different masses and energies flowing around here You want to say at this point? This is what geometry looks like and if you're working near that point Then you may as well be working in a little copy of Euclidean plane and over here. There's another notion of geometry and Here there's another notion of geometry So what we need to do is Say for each position P in space there is a Geometry there and Each geometry is essentially the same in the sense that it looks like Euclidean geometry at that point however That geometry can vary from place to place if you in at some locations it's easier to move one direction versus a different direction and The notion of which direction is biased say which direction is attractive due to gravity Versus moving across the gravitational field lines that can vary from place to place so what we need is a Hermitian or actually since working with the real numbers here a symmetric Matrix at every point And we call this G of P. So P is the point you're sitting at this is P and Nearby P. There's a matrix G a symmetric matrix Which we can stir the geometry given by VW Defined by G and G is called the metric So what the spectral theorem shows us is if you can define at every point a symmetric matrix a positive definite symmetric matrix in fact then at every point in space if You use a basis at that point Then in the correct or orthogonal basis at that point you may as well be working in the traditional Euclidean metric And this is why I say if you're standing in one spot, you can't really tell that you're living in a wobbly stretch of universe However, if you look across very different positions and compare them back and forth you realize That the way you measure things over here is different than the way you measure things over there And that's saying that our notion of geometry even though at each point it looks like Euclidean space Can vary from place to place and the spectral theorem allows us to do this So again the idea is we have a second factor function From space so G is a function from space To the space of symmetric matrices and G of P Gives you the geometry at point P Using the inner product defined by that particular metric G Now of course this is a course in scientific computing not a course in general relativity But this idea of adapting the geometry to the point or to the situation using a rotation And using and then analyzing how much things are stretched in different directions This is going to be a key of a lot of computational maneuvers And it just so happens it's also the fundamental fact that lets our universe Make sense mathematically in terms of geometry and gravity