 We are discussing the notion of isomorphisms on inner product spaces okay so an isomorphism on an inner product space let us recall must be a vector space isomorphism which also preserves the inner product okay and we had seen last time that a linear transformation preserves inner products if and only if it preserves a norm okay and we had also seen a characterization of isomorphisms on inner product spaces okay. Let us discuss the notion of unitary operators next unitary operators I remember having given the definition but I will recall this definition a unitary operator on a vector space on an inner product space a unitary operator on an inner product space is an isomorphism is an isomorphism of the space onto itself it is an isomorphism of this as an inner product space so it is a vector space isomorphism and preserves inner products so that is the definition of a unitary operator we will look at one or two results for unitary operator some examples and then for a linear transformation what happens to the matrix of the linear transformation when you change from one basis to another this has been studied earlier we look at what happens when the bases are orthonormal bases okay what are the properties of unitary operators suppose you have let u1, u2 be unitary operators on an inner product space v let me say on an inner product space v look at the okay then look at the product we will show that the product is also unitary for one thing the product u1, u2 is invertible see the claim that I am trying to make here is that if u1 and u2 are unitary then I want to show that the product is also unitary now what is a unitary map it must be an isomorphism so it must be a vector space isomorphism and must preserve inner products for the product to be a vector space isomorphism it must be invertible but this is invertible because u1 and u2 are unitary so they are invertible the product is invertible in fact there is a formula for the inverse also u1, u2 inverse is u2 inverse u1 inverse so the product is invertible I will next show that the product preserves inner products then it will mean that this product means composition of unitary maps also look at norm of I want to show that u1, u2 preserves inner products but we have seen that a mapping preserves inner products on a vectors on a inner product space if and only if it preserves norms so I will simply show u1, u2 preserves norms so look at u1, u2, x this is u1 of u2, x so since u1 is unitary it preserves a norm it preserves inner products so it preserves norm so this must be norm u2, x because u1 is unitary again u2 is unitary so this must be norm x and so you have shown norm u1, u2, x is norm x for all x and so u1, u2 preserves a norm so it must preserve the inner product this is the norm induced by the inner product okay so the product is so u1, u2 is unitary so the set of unitary operators on a vector space on an inner product space it is close with respect to composition what about inverse is the inverse also unitary let u be unitary then u inverse exists because it is an isomorphism then u inverse exists we want to show that u inverse preserves norm so you consider norm of okay consider y to be u inverse x composition these are linear transformations so whenever I write this it is a composition when I write down the matrix of these transformations then it is a matrix product so here it is a composition operation I want to show that u inverse preserves norm so it will follow that u inverse preserves inner products since u inverse inverse exists it will follow that it will then follow that u is a unitary operator so consider y equal to u inverse x I want to show u inverse preserves norm this means x equals u y I want to show norm u inverse x equals norm x okay then we have norm u inverse x equals norm y but norm y is norm u y because u is unitary but u y is x so this is true for all x so I have shown that norm u inverse x equal to norm x u inverse is of course invertible u inverse inverse is u so it is a vector space isomorphism that preserves inner products that preserves norms so u inverse is unitary what about the identity operator is that unitary obviously no identity is unitary I the identity operator is obviously unitary so what follows is that the set of all unitary operators on inner product space forms a group the set of all the unitary operators on an inner product space is a group is a is a multiplicative group where the multiplication is defined as a composition of the operators so this is a multiplicative group of course it is not necessarily abelian okay remember this result that we proved last time if T is a linear transformation from V into W dimension of V is equal to dimension of W V and W are of course inner product spaces then T preserves inner products it is the same as saying that T is a inner product space isomorphism that is the same as saying that T takes an orthonormal basis to an orthonormal basis which is the same as saying that T takes every orthonormal basis to an orthonormal basis onto okay so what follows is that if U is a unitary operator just an operator if U is an operator on an inner product space finite dimensional if U is an operator on a finite dimensional inner product space then U is unitary if and only if U preserves inner products so let me just recall what it means when we say that an operator preserves inner product inner product U X U Y is equal to inner product X Y this must be true for all X Y okay this follows easily U is an operator on a finite dimensional vector space now dimensions coincide then U is unitary if and only if U preserves inner products this follows from that theorem okay. Let us also recollect the notion of the adjoint of an operator okay the adjoint of an operator we proved in the case of finite dimensional space okay there is a characterization of unitary operators in terms of the adjoint operation so that is what I want to discuss next so this next result holds in general in an infinite dimensional space okay so the result is dimension free. So what I am trying to say is that there is a connection between the unitary operators on a general vector space and the adjoint operation so that is the following so I will prove this as a theorem let V be an inner product space U be an operator on V linear operator on V then U is unitary U is unitary if and only if U star U equals U U star equals I the identity operator on U V is an inner product space not necessarily finite dimensional in order to see remember we have shown that the adjoint operator of a linear operator exists on a finite dimensional space okay so we are already talking about the and remember that in general the adjoint operator over an infinite dimensional space does not exist okay but we are already talking about adjoint operator on a possibly infinite dimensional inner product space that is possible if the operator is unitary that is what this result says okay proof suppose U is unitary there are two implications right so let us prove the first part necessity suppose that U is unitary then what is the definition of a unitary operator it is an isomorphism of the inner product space on to itself so in the first place it is an invertible linear transformation so U inverse exists we must show that U preserves inner products then it follows that U is then it follows I am sorry given U is unitary we want to show that this identity holds now these two equations hold now these two equations can you figure out that it is the same as saying that U inverse equal to U star okay so we will just show that U inverse is equal to U star U inverse exists the claim is that U inverse equal to U star okay so you consider I want to show U inverse is U star okay consider U inverse X, Y I can write this as U inverse X, let us see U U inverse Y now I want to show U star U equal to identity what is given is that U is unitary so this is U star is adjoint star is always adjoint operator you first agree with this I have just applied the fact that U inverse exists so Y is written as U U inverse Y so but okay I need to show U inverse star but okay U inverse star is U star inverse let us see this is X, U inverse star U inverse Y this is X, U inverse star is U star inverse I think it is not it is not easy I mean I could have I can write U star inverse but first of all I want to show that star exists okay let us see I need to give a different proof I want to show U inverse is U star and write it as okay I should have started with inner product U X, Y yes that is the same as U X, U inverse Y okay that is the same as U X, U inverse Y U is unitary so this U kind of can be removed this is X, U inverse Y this is X, U inverse Y so what this means is that by the definition U star is that unique operator which satisfies the equation U X, Y equals X, U star Y but what we have shown is that U X, Y is X, U inverse Y so it means that U star must be U inverse so if U is unitary then U star is equal to U inverse so these two equations hold trivial in that case that is the first part sufficiency if U satisfies these two equations then I want to show that U is unitary for one thing U is invertible if U U star equals U star U equals identity then it follows that U inverse equals U star so U is invertible I will simply show U preserves inner products okay that we prove that U is unitary so it is invertible I need to show that it preserves inner products so consider U X, U Y this is X, so what we know is that U star is U inverse so I will first write X U star U Y U star is U inverse so this is just X, Y so U preserves inner products U is invertible and U preserves inner products so U is unitary okay so it is intimately connected to the operation of taking the adjoint U star, U star exists and U star is equal to U inverse for a unitary operator. The original definition is just an isomorphism of an inner product space on to itself isomorphism over vector spaces plus the factor it preserves inner products okay so this is dimension free okay. Before I look at examples I want to look at the matrix case that is we have it is really a little theorem over finite dimensional spaces what happens to the matrix of a unitary operator with respect to an orthonormal basis let U be unitary over V with V being finite dimensional the matrix of U denoted by A relative to any means every orthonormal basis the matrix of U denoted by A so A is a matrix of U relative to an orthonormal basis satisfies A star A equals identity what will also follow is that this is equal to A A star see this is like a square matrix acting on a finite dimensional space so if it has a left inverse then it must have a right inverse and those two must coincide okay so the last part comes from that result the converse is also true the converse also holds what is the converse you take a matrix that satisfies the equation A star A equal to I and look at a unitary operator which satisfies a property that there is an orthonormal basis such that the matrix of this U with respect to that orthonormal basis is A then that U must be unitary okay that is a converse. Take a matrix A that satisfies this equation and then look at the unitary operator it can be this can be easily defined all that you have to do is look at the operator U that satisfies U X equals A times X this U has a property that the matrix of U relative to some basis will be equal to A then you can show that the unit you can show that the operator U is unitary okay so that is the converse so proof by the way an operator that satisfies this equation is called I am sorry a matrix that satisfies this equation is called a unitary matrix matrix coming from matrix coming from a unitary operator relative to an orthonormal basis is called a unitary matrix if the space is a real it is called an orthogonal matrix the space is a real space then it is called an orthogonal matrix okay. I want to show that this is satisfied converse, converse is also easy, okay. First part I must show A star A equals identity, let us take a basis script B be an ordered orthonormal basis relative to any ordered orthonormal basis, our basis will always be ordered. Let B equal, let me use U1, U2, etc. UN, let B be an ordered orthonormal basis of the vector space V, call the matrix of U, use the unitary operator that is given to me, call the matrix of U relative to this basis as A, we must show that this A satisfies this equation, okay. Look at A star A, this is a matrix product, A star is the matrix of U relative to B star into the matrix of U relative to B, this is matrix multiplication but this is we have seen last time that this star can be taken inside, it is U star relative to B into U relative to B, the matrix of U relative to B. This is the product of two matrices, this will correspond to the composition, this will correspond to the matrix of the composition of U star U which we have done even for finite dimensional space, there is no inner product involved here, okay. But then U star U is unitary so U star U is identity, we have proved it last time, so this is identity relative to B, identity relative to B simply means identity with respect to BB, right that is in our notation, it is the same basis with respect to which we are doing this, which is the identity matrix, okay. Remember the notation that we have employed, in general we talk about the matrix of T relative to two basis B1, B2, when B1 is equal to, when T is an inner operator on a vector space then it makes sense to use only one basis, in that case B1 equals B2, in that case we will simply say TB, so that is the notation I am using here. Identity B means identity, you write every basis element in terms of the basis element and write down the matrix that is the identity matrix, so A star equal to I that proves the first part, the matrix of U relative to any orthonormal basis will satisfy the equation A star equal to identity. Conversely suppose that A star A equals I and U is the matrix, U is the matrix of A relative to some basis, A is the matrix of U relative to some basis, okay. A being the matrix of U relative to an orthonormal basis that is U X equals A X corresponding to that basis in which one first part, U star U equal to identity using orthonormal basis I do not think I have used, there is only one property that I have used that this product will be this one, otherwise I have not used orthonormal basis, there is one place where we have used in writing this down. This is not true in general, please observe this is not true in general because say I have made this remark I remember the time when we discussed I think I called it A and B, if A is the matrix of U relative to an orthonormal basis B and B is the matrix of U star relative to the same orthonormal basis then A is equal to B star. If you use that is this is a very simple relationship between the matrices of U and U star corresponding to the same orthonormal basis, if you do not take an orthonormal basis then this relationship does not hold anymore so that is where it is being used. For this you do not need unitary, this is true for any operator, see this relationship is true for any operator provided B is an orthonormal basis that is correct, U star U is a unitary is being used here when I wrote down U star you equal to identity, see this identity transformation inside is identity transformation that is because U is unitary yes so it is being used here that is you can take a simple example of a 2 by 2 or a 3 by 3 vector space under transformation over that and then determine the matrix of U relative to an ordinary basis compute U star write down the matrix of U star relative to this ordinary basis not necessarily orthonormal then they are not related like this okay so that is where this is being used okay the converse, converse this matrix A is the matrix relative to it is a matrix of U relative to some orthonormal basis it satisfies this equation I must verify that U is unitary okay. So U matrix of U relative to that orthonormal basis U relative to an orthonormal basis I will call it B relative to the orthonormal basis which I will call script B so the matrix of U relative to B is A by definition now look at A star A this is equal to identity since A star equal to identity we have A star is the matrix of U relative to B star U B equals identity matrix left hand but identity matrix is as I have written there it is identity operator relative to the same basis okay which the short notation is identity operator relative to just this basis remember this that if you take the identity operator and then take two basis then the matrix will not be identity okay the matrix of the identity linear transmission with respect to two different basis will not be identity but what is important is that the matrix will be invertible if two basis are the same then the matrix of the identity transformation relative to these two basis is the identity matrix okay. So this right hand side is identity matrix on the left hand side I have again B is an orthonormal basis so the star can be taken inside it is U star U relative to B that is equal to identity relative to B see both sides I have matrices but what is important is this identity operator inside this is the operator product U star U operator composition U star U so can you see from this that there is a basis what is the meaning of this statement the meaning of this statement is that if just write it elaborately if B is let us call U 1 U 2 etc U n then this equation tells me that U star U star U of U 1 is not a very good notation let us say W 1 W 2 etc W n then U star U W 1 must be equal to W 1 because of this etc U star U of W n must be equal to W n this happens if this happens then this must hold otherwise you would not get this right if this does not happen this is not true so from this it this is a basis W 1 etc W n is a basis so it means from this U star U star U star U star U star U star U star U star University operator the same thing you can do for U U star to conclude that U star equals identity so what we have proved is is u is unitary okay. Composition, but that is not the definition u star of u of x is not u star of x into u of x in fact there is no meaning to that u star x into u x means what? u star x is a vector u x is another vector so into means what? The multiplication means what? There is no meaning to that it is always a composition, it is always a composition operation this is the composition operation of always okay and see there is no complex number involved it is just a adjoint operator and it is a composition of the adjoint with the operator u. So there is no complex number involved here these are equations involving operators u star u is an operator defined by u star u of x equals u star operating on u of x okay. So this is what we have then the matrix of a unitary operator relative to an orthonormal basis satisfies the equation A star A equal to A star equal to identity the converse is also true okay. Let us now look at some examples okay before that I will just put down this definition A in C n cross n is called unitary if A star A equals identity so this is just for matrices unitary matrix A element of R n cross n is called orthogonal if A transpose A equals identity that is if the space is real space and if you take a unitary operator and write down the matrix of the unitary operator relative to any basis then let us call it A then A star is actually A transpose real space A star is A transpose so this will hold such matrix such matrices are called orthogonal matrices so when I say orthogonal matrix it is a real space in general unitary means it is a complex space. So let us look at some examples examples of unitary matrices the one dimensional case A is 1 cross 1 so there is just one vector C A is just this then the real case A is orthogonal if and only if C equal to plus minus 1 A is unitary if and only if what is the adjoint in that case what is the adjoint of see what is C star when C is a complex number just C bar no so if and only if C bar C is 1 which is true if and only if mod C is 1 that is if and only if C is e power i theta theta real C is e power i theta where theta is a real number so one dimensional case is easy two dimensional case let us look at the orthogonal case unitary I will leave it as an exercise example 2 take the case A B C D I am looking at the real case I want to determine conditions on A B C D what is the form of the matrix A if it is orthogonal that is a real case A is orthogonal if and only if A transpose equals A inverse similar to U star equals U inverse for the unitary case what is A inverse A inverse you must say I am trying to determine conditions under which A is orthogonal so A inverse exists 1 by ad minus B C into D A minus B minus C that is A inverse right but what is A transpose on the other hand A transpose is A C B D so these two matrices must coincide so A must be of the form can tell me by the way what is the ad minus B C the determinant of an orthogonal matrix so let me say also determinant of A if it is orthogonal plus or minus 1 so this ad minus B C is plus or minus 1 so that leads to the following two cases if it is plus 1 minus 1 those are the two cases I will write down those two expressions you must compare this with this this is equal to that with ad minus B C equal to 1 this is equal to that with ad minus B C equal to minus 1 so A has either this form just check this B minus B or can just fill up the other two entries is that okay in the first case D is equal to A in the case when ad minus B C is 1 D is equal to A B is equal to minus C so when I write B here I get a minus B here in the case when this is minus 1 A is D is minus A and so I get this and these two must have the same sign B and minus B so this is minus A square minus B square that is a determinant that is minus 1 minus 1 whereas this ad minus B is in the second case is also minus 1 okay so please check these calculations what is the corresponding result for complex case that will be an exercise A element of C 3 cross sorry 2 cross 2 implies A is what a similar analysis can be carried out let us also look at one more example look at the vector space C n cross n with the inner product inner product of A B is trace of A B star we have encountered this before we have also encountered the operator left multiplication L M of A is equal to M times A for a fixed M left multiplication by the matrix M then we have computed the adjoint of this so what is that left multiplication by M star okay that is we have computed that L M star of A is M star of A M star into A that is L M star is L M star for this operator we have L M is unitary if and only if can you make a guess M star is unit M is unitary L M is unitary if and only if M is unitary okay let me prove one part I will leave the other part for you let me take the case when M is unitary prove L M is unitary okay so I will just prove one part prove of sufficiency M star M equals M M star equals identity I want to show L L M is unitary first L M is invertible consider L M of A to be the 0 matrix this means M A is a 0 matrix but M is unitary in particular M is invertible so this means A is 0 so the null this is linear the linear map of course so the null space of L M is single term 0 so L M is invertible because it is on the same space finite dimensional so L M is invertible I must show that L M preserves in a product then it would follow L M is unitary so consider L M A L M B in a product L M by definition is M star A L M B is M star B this is I will without appealing to this I will straight away okay let us appeal to that so we need to I can actually write this as that means verification I can write this is equal to A comma M M star B M M star is identity so it is just A comma B okay okay that means I leave this as an exercise I will still proceed this is equal to this needs a little justification but I am leaving that as an exercise again this is A comma M M star B but M M star is identity so this is A comma B and so L M preserves in a products it is M A comma M B M star M yes M A comma M B and then you can push this M star that is needs a proof but that can be done okay so this is another example so there are two parts that you need to verify one is this equation the other one is the necessary part if L M is unitary show that M is unitary it is almost time okay maybe I will stop what I want to discuss next is how are the this question I address in the beginning also given a matrix of a linear transformation corresponding to two bases we know that they are related by the formula that is let us say I have B as the matrix of A relative to a basis B 2 A is a matrix of matrix of the transformation T relative to B 1 then B is equal to M A sorry that is correct M A M inverse right we have we have proved this before what happens in the case of orthonormal basis there is something more that we can say for the invertible matrix M that will turn out to be unitary in the ordinary basis case M is invertible in the case of orthonormal basis M will turn out to be a unitary matrix okay we will prove this result and then consider the notion of self-adjoint operators normal operators next time okay I will stop.