Inner Product for Complex Vector Spaces

we spoke last about inner products as a structure that we are to a vector space and in particular we Illustrated this with the case of a real vector space but as you know already in quantum mechanics we need complete vector spaces so now let's consider what changes if you have a complex vector space so inner products on complex vector spaces vector spaces now we're not going to use anymore this dot notation that we use for the real case but rather a notation that this a little more involved but shows the physics more clearly and the notation is that it's going to show that again as in the case of the real vector spaces you start with two vectors and you get a number this time you start with two vectors in the complex vector space and you will get a number that could be complex as well so this will be denoted by an object like this in which you put one vector here and one vector here and one vector there so there will be two inputs I left input on the right input recoverin with the dot products we do a dot B and we happen to say that was equal to B dot a but now we're going to put this two inputs like that so one vector here one vector there this object called the inner product is going to give you a number so it's going to be a complex number so this is our inner product and the two vectors are going to be put first here second there may make a difference where you put them so the inner product is a machine that takes two vectors and gives you a number so for inspiration we can try to think of what we would do in a complex vector space complex numbers to the nth power this is a vector space where you have V 1 Z 2 up to V M n entries and those are the vectors now suppose you had of this vector and you wanted to figure out what is the length square of it that's a vector and you're going to get a number you're familiar with complex numbers what you do is define the length of well if you had a single a complex number you know what you're supposed to do to get the length of this complex number you multiply by its complex conjugate that's the length squared for a collection of components each of which is a complex number you would do the following of the norm squared of this whole thing that was called V would be v1 z1 and you know you should star 1 plus Z 2 3 2 star V n and this has the good property that this thing is a real number and is positive and in fact it vanishes if and only if all this complex numbers vanish so this is a nice model for a length squared and it suggests also a nice possible definition of an inner product in which if you have another complex vector in this n-dimensional complex vector space you could say maybe I should use for wze I should use W 1 star z1 plus all the way up to W n star Z n and if this is the case notice that W and V are treated very differently of the W's are complex conjugates and disease are not and this would be necessary so that then because identify this length squared as nothing but the inner product of Z with Z because if you use that evolute V with V use in this formula you get exactly this so this is a very reasonable way of thinking about what could be an inner product in a complex vector space but again as we want to make sure here our important step is to get from these things a set of axioms that without telling me exactly what's the formula for the inner product it tells me all the things that are necessary for this inner product to be useful and that's what we want of course to understand so here are the axioms that we should put axioms for and the first axiom is going to be very similar the first few axioms are going to be very similar to the ones we had in the real case first axiom is that V V is greater or equal than 0 so put two vectors on the inner product the same vector here and then you will get something greater than or equal to zero in general in particular this thing is real you see this come to this inner product be a complex number in general it is but if you put the same one it's implicit here because a complex number we don't say is greater than zero this is a real number so I'll mention here V V is real the second axiom will be that of V V equals zero if and only if the vector is zero so it's equivalent to V zero and that is intuitively clear here in this example and follows it it's a nice thing a very useful property and it follows from well it doesn't follow it it is used to prove many useful properties when you have to show some operators are equal to another operator some vectors equal to another vector you calculate the difference between the two vectors and show that their norm is 0 therefore vector is 0 so you will see a lot of that let's continue there is a two more properties V or let me stay the way I have it here you will love with V 1 and V 2 you V 1 plus V 2 is equal to u V 1 plus U V 2 then you can get a number u alpha v where alpha is a number the number goes out alpha u v and finally the exchange property cannot be as simple as you V is equal to V u that's not quite true cannot be it's not satisfied by this prop this example and it's no good so the property that you need here is this is complex conjugated so if you change the order in the inner product you get the complex conjugate number and this is a fundamental property that makes it different from the case of the real vector space in some sense these definitions are good also for the real case where you say welcome plus contradiction of a real number doesn't change a real number so nothing is changed but this makes a world of difference and it's clearly satisfied by this property if you had put this Z to the left and the W to the right disease would have been complex conjugated than the W is not which is a result of complex conjugate in this right hand side so these are our axioms and let's try to understand a little better but they say a couple of simple properties of comment you with zero the inner product of a vector with zero is zero you could show it by putting V 2 being equal to zero here and that way you have here V 2 equal to 0 and if we have 0 plus V 1 is V 1 so if you V 1 to V 1 they cancel and then you get u 0 so this is 0 follow from the axioms V Rho u well V Rho U is the complex conjugate of U 0 and that's the complex conjugate of the 0 vector and that's the 0 vector no this is zero number I'm sorry this is the zero number this is the zero number the complex conjugate of the zero number is the zero number so those are simple properties now notice this two properties here this are the properties of linearity in the second argument you can do anything linear in the second and when you put one plus another vector it's linear in that it's linear here you can ask why didn't you state what happens if you put two a sum of two vectors here or a number times a vector there and the reason we don't state that is because it can be calculated so even though we have fully been arity full linearity on the second argument second entry let's look at what we have on the first entry so suppose I wanted to compute based on that thing u1 plus u2 comma V well now you can imagine you use the fifth property and said well that's a complex conjugate of V u 1 u 2 Plus u 2 well but here I can use the axiom to expand so it's a complex conjugate of V u 1 plus V u 2 and the complex conjugate of a sum is the sum of complex conjugates of course so this is V 1 u 1 complex conjugate plus V it's not V 1 is V u 2 complex conjugate and the complex conjugate of an inner product is the one reverse so you get you one v+ you to be so look are you can expand on the first entry just the same way you could expand on the second entry so the so far nothing too unusual going on here but now let's try the other property what happens if you have a constant so if I have a constant here alpha u V this would be equal to the complex conjugate of V alpha u now the Alpha can go out so you have alpha v u complex conjugate alpha gets complex conjugated this is a product of two complex numbers and the second gets complex conjugated but the complex conjugate of V U is U V so here you get something quite entertaining and very easy to forget and make mistakes conjugate homogeneity on this first entry on the entry to the left any complex number goes out of the bracket with a complex conjugate so that shows that the role of the first entry in the second entry is not identical and we call this conjugate it homogeneity very important property that you should be aware of so with inner produce here and once you have an inner product its natural to say it is something I'm sure you have seen before that we call two vectors orthogonal if their inner product vanishes so of U and V are orthogonal V are orthogonal when u v is equal to zero and that of course means also that V U is zero so now at symmetry so you don't have to say that these vectors orthogonal to the second one but the second is not orthogonal to the first there are mutually orthogonal if this is our seventh of these this is orthogonal to that some mutual relation the order here does not matter because you're setting it equal to zero so this equation implies this so those are two vectors that are orthogonal now this inner product is said to be also non degenerate non degenerate is a very nice property it says that the following doesn't happen you cannot have a vector that is non zero and has zero inner product with everybody you just can't find such vector that doesn't happen the inner product is non-degenerate and this is supposed to mean that if you have a vector that has zero inner power with everything else the vector is zero so if is non-degenerate if the vector are X belonging to V is such that X with V is equal to 0 for all V then X is equal to 0 so if you have a vector whose inner product with V is 0 for all V then the vector is 0 and the proof is just far too simple it's almost an obvious statement if this is true then you can take V to be x in which case you would have so proof take V equal X and therefore you will have X x equals 0 and you know by our postulate here that are our axiom that V V equals 0 means V equal to 0 that this X x equals 0 implies x equals 0 so the inner product is non-degenerate um two more properties hold that are interesting the Schwarz inequality still calls inequality so how do we write it you V is less than or equals you V so this is the quality you proved it or you saw the proof in the last video for the case of real vector spaces now for the case of complex vector spaces you will see this in the exercises it's an important property used all the time use for the uncertainty principle for well you said many many times in this course and look what everything means here maybe I have not been totally precise here oh I probably should add here that the length squared or norm squared of any vector is defined to be V so the Schwarz inequality is this thing here is the inner product of two vectors there is a complex number this thing the notes here the absolute value of a complex number in here you have a vector and this is the norm of the vector the norm of this vector as in this equation so V absolute value of a vector is the norm of a vector so this is the Schwarz inequality and it's saturated that is to say it becomes an equality saturated when V is a constant times U and this is parallel V is parallel to you for complex vectors this constant could be a complex number so that's the Schwarz inequality and we'll talk a lot more about it there's also triangle inequality angle inequality and this is to say that u plus V is less than or equal than u plus B it's another nice property and geometrically something it is quite simple you've seen it a vector u a vector V the vector u plus V the little u plus v s-- a little less long than the sum of the lengths of the vectors 1 & 2 so when is this saturated if u saturated when armed yeah you probably could imagine when this is saturated it's a little different than here we fade it V is equal to C U and here C was just a complex number here is a little more arm subtle and you could imagine that if you think of this complex for complex numbers you have a first complex number and a second complex number when is it that the sum of the two complex numbers is equal to the sum of the length of this one well they have to be parallel but moreover they have to point in the same direction or whether they would subtract and the length of the sum would be smaller than the separate sum of the lengths so in this case it's necessary not only that the vectors be parallel but the concept of proportionality must be a real number and must be positive so we just say positive because we use the word two different negative only for real numbers so this is the saturation of the short inequality so we got now our complex vector spaces with of an inner product so in a sense we've gotten to a very important point because this is when the vector space is finite dimensional the definition of a Hilbert space so a finite dimensional hilbert space is a finite dimensional complex vector space with an inner product that satisfies all these properties so here it is Hilbert spaces for you infinite dimensional case everything again must be the same you must have an infinite dimensional complex vector space and you must have this axioms but there's one more axiom that must be true for the infinite dimensional hilbert space to be for the infinite dimensional complex space to be a Hilbert space and that's a more subtle condition of sequences of vectors so if you have a sequence of vectors infinite sequence of vectors on a complex vector space it must be true that the limit of that sequence of vectors when it exists is in the vector space you cannot remove some singular point this is a subtle condition more precisely stated every Cauchy sequence of vectors must converge to a vector in the space now we say because it's important and someday you may be working with a complex vector space infinite dimensional one that this subtle and things are confusing and convergence properties are not clear and this may be necessary but at least for the purposes of our course even though infinite dimensional complex vector spaces will be there all the time we will have no particular use for that

Channel: Jacob Bains

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Id: Kr3X1Pa9N3E

Length: 25min 56sec (1556 seconds)

Published: Fri Apr 28 2017