Spectral Theorems on Euclidean Spaces
In Chap.7we studied the operation of changing a basis for a real vector space. In particular, in the Theorem7.9.6and the Remark7.9.7there, we showed that any matrix giving a change of basis for the vector spaceRnis an invertiblen×nmatrix, and noticed that anyn×ninvertible yields a change of basis forRn.
In this chapter we shall consider the endomorphisms of the euclidean space En = (Rn,·), where the symbol ·denotes the euclidean scalar product, that we have described in Chap.3.
152 10 Spectral Theorems on Euclidean Spaces
form an orthonormal basis forE2, the matrix A = √12
1 1 1−1
is orthogonal.
Proposition 10.1.3 A matrix A is orthogonal if and only if
tA A = In, that is if and only if A−1 = tA.
Proof With(v1, . . . , vn)a system of vectors in En, we denote by A = (v1· · ·vn) the matrix with columns given by the given vectors, and by
tA =
⎛
⎜⎝
tv1
...
tvn
⎞
⎟⎠
its transpose. We have the following equivalences. The matrixAis orthogonal (by definition) if and only if(v1, . . . , vn)is an orthonormal basis forEn, that is if and only ifvi·vj =δi j for anyi,j. Recalling the representation of the row by column product of matrices, one hasvi·vj =δi j if and only if(tA A)i j =δi j for anyi,j,
which amounts to say thattA A=In.
Exercise 10.1.4 For the matrixAconsidered in the Exercise10.1.2one has easily compute thatA = tAandA2 = I2.
Exercise 10.1.5 The matrix
A = 1 0
1 1
is not orthogonal, since
tA A = 1 1
0 1 1 0 1 1
= 2 1
1 1
= I2.
Proposition 10.1.6 If A is orthogonal, thendet(A) = ±1.
Proof This statement easily follows from the Binet Theorem5.1.16: withtA A=In, one has
det(tA)det(A) = det(In) = 1,
and the Corollary5.1.12, that is det(tA)=det(A), which then implies
(det(A))2=1.
Remark 10.1.7 The converse to this statement does not hold. The matrixAfrom the Exercise10.1.5is not orthogonal, while det(A)=1.
Definition 10.1.8 An orthogonal matrixAwith det(A)=1 is calledspecial orthog- onal.
Proposition 10.1.9 The setO(n)of orthogonal matrices inRn,n is a group, with respect to the usual matrix product. Its subsetSO(n) = {A ∈ O(n) : det(A)=1} is a subgroup ofO(n)with respect to the same product.
Proof We prove that O(n)is stable under the matrix product, has an identity element, and the inverse of an orthogonal matrix is orthogonal as well.
• The identity matrixInis orthogonal, as we already mentioned.
• IfAandBare orthogonal, then we can write
t(A B)A B= tBtA A B
= tB InB
= tB B = In, that is,A Bis orthogonal.
• IfAis orthogonal,tA A=In, then
t(A−1)A−1 = (AtA)−1 = In, that proves thatA−1is orthogonal.
From the Binet theorem it easily follows that the set of special orthogonal matrices is stable under the product, and the inverse of a special orthogonal matrix is special
orthogonal.
Definition 10.1.10 The group O(n)is called theorthogonal groupof ordern, its subset SO(n)is called thespecial orthogonal groupof ordern.
We know from the Definition10.1.1 that a matrix is orthogonal if and only if it is the matrix of the change of basis between the canonical basis E (which is orthonormal) and a second orthonormal basisB. A matrixA is then orthogonal if and only ifA−1 = tA(Proposition10.1.3).
The next theorem shows that we do not need the canonical basis. If one defines a matrixAto be orthogonal by the condition A−1 = tA, then Ais the matrix for a change between two orthonormal bases and viceversa, any matrix A giving the change between orthonormal bases satisfies the conditionA−1 = tA.
Theorem 10.1.11 LetCbe an orthonormal basis for the euclidean vector space En, withBanother (arbitrary) basis for it. The matrix MC,Bof the change of basis from CtoBis orthogonal if and only if also the basisBis orthonormal.
154 10 Spectral Theorems on Euclidean Spaces Proof We start by noticing that, sinceCis an orthonormal basis, the matrix ME,C giving the change of basis between the canonical basisEandCis orthogonal by the Definition10.1.1. It follows that, being O(n)a group, the inverseMC,E = (ME,C)−1 is orthogonal. WithBan arbitrary basis, from the Theorem7.9.9we can write
MC,B= MC,EME,EME,B
= MC,EInME,B = MC,EME,B.
Firstly, let us assumeBto be orthonormal. We have then thatME,B is orthogonal;
thusMC,Bis orthogonal since it is the product of orthogonal matrices.
Next, let us assume that MC,Bis orthogonal; from the chain relations displayed above we have
ME,B = (MC,E)−1MC,B = ME,CMC,B.
This matrixME,Bis then orthogonal (being the product of orthogonal matrices), and
thereforeBis an orthonormal basis.
We pass to endomorphisms corresponding to orthogonal matrices. We start by recalling, from the Definition3.1.4, that a scalar product has a ‘canonical’ form when it is given with respect to orthonormal bases.
Remark 10.1.12 Let C be an orthonormal basis for the euclidean space En. If v,w ∈ En are given by v=(x1, . . . ,xn)C and w=(y1, . . . ,yn)C, one has that v·w=x1y1+ · · · +xnyn. By denoting X andY the one-column matrices whose entries are the components ofv,wwith respect toC, that is
X =
⎛
⎜⎝ x1
...
xn
⎞
⎟⎠, Y =
⎛
⎜⎝ y1
...
yn
⎞
⎟⎠,
we can write
v·w = x1y1+ · · · +xnyn =
x1 . . .xn
⎛
⎜⎝ y1
...
yn
⎞
⎟⎠ = tX Y.
Theorem 10.1.13 Letφ ∈ End(En), withEthe canonical basis of En. The follow- ing statements are equivalent:
(i) The matrix A=MφE,Eis orthogonal.
(ii) It holds thatφ(v)·φ(w)=v·w for anyv,w∈ En.
(iii) IfB = (b1, . . . ,bn)is an orthonormal basis for En, then the setB =(φ(b1), . . . , φ(bn))is such.
Proof (i) ⇒(ii): by denotingX =tvandY =twwe can write
v·w= tX Y, φ(v)·φ(w)= t(A X)(AY)=tX(tA A)Y,
and since Ais orthogonal,tA A=In, we conclude thatφ(v)·φ(w)=v·wfor anyv,w∈En.
(ii) ⇒(iii): let A=MφC,C be the matrix of the endomorphismφwith respect to the basisC. We start by proving that Ais invertible. By adopting the notation used above, we can represent the conditionφ(v)·φ(w) = v·was t(A X)(AY)=
tX Y for any X,Y ∈ En. It follows thattA A=In, that isAis orthogonal, and then invertible. This means (see Theorem7.8.4) thatφis an isomorphism, so it maps a basis forEn into a basis for En. IfBis an orthonormal basis, then we can write
φ(bi)·φ(bj) = bi·bj = δi j
which proves thatBis an orthonormal basis.
(iii) ⇒(i): sinceE, the canonical basis for En, is orthonormal, then(φ(e1), . . . , φ(en))is orthonormal. Recall the Remark7.1.10: the components with respect toEof the elementsφ(ei)are the column vectors of the matrixMφE,E, thusMφE,E
is orthogonal.
We have seen that, if the action ofφ∈End(En)is represented with respect to the canonical basis by an orthogonal matrix, thenφis an isomorphism and preserves the scalar product, that is, for anyv,w ∈ Enone has that,
v·w=φ(v)·φ(w).
The next result is therefore evident.
Corollary 10.1.14 Ifφ∈End(En)is an endomorphism of the euclidean space En whose corresponding matrix with respect to the canonical basis is orthogonal then φpreserves the norms, that is, for anyv ∈ Enone has
φ(v) = v.
This is the reason why such an endomorphism is also called anisometry.
The analysis we developed so far allows us to introduce the following definition, which will be more extensively scrutinised when dealing with rotations maps.
Definition 10.1.15 Ifφ∈End(En)takes the orthonormal basisB=(b1, . . . ,bn) to the orthonormal basisB=(b1 =φ(b1), . . . ,bn=φ(bn))in En, we say thatB andBhave the sameorientationif the matrix representing the endomorphismφis special orthogonal.
156 10 Spectral Theorems on Euclidean Spaces Remark 10.1.16 It is evident that this definition provides an equivalence relation within the collection of all orthonormal bases for En. The corresponding quotient can be labelled by the values of the determinant of the orthogonal map giving the change of basis, that is detφ= {±1}. This is usually referred to by saying that the euclidean spaceEnhas two orientations.