12.3. RIESZ REPRESENTATION THEOREM, ADJOINT MAP 315

Definition 12.3.3 The linear map, A∗ is called the adjoint of A. In the case when A : X→ Xand A = A∗, A is called a self adjoint map. Such a map is also called Hermitian.

Theorem 12.3.4 Let M be an m× n matrix. Then M∗ =(M)T in words, the transpose of

the conjugate of M is equal to the adjoint.

Proof: Using the definition of the inner product in Cn,

(Mx,y) = (x,M∗y)≡∑i

xi∑j(M∗)i j y j = ∑

i, j(M∗)i jy jxi.

Also (Mx,y) = ∑ j ∑i M jiy jxi. Since x,y are arbitrary vectors, it follows that M ji = (M∗)i jand so, taking conjugates of both sides, M∗i j = M ji ■

Some linear transformations preserve distance. Something special can be asserted aboutthese which is in the next lemma.

Lemma 12.3.5 Suppose R ∈ L (X ,Y ) where X ,Y are inner product spaces and R pre-serves distances. Then R∗R = I.

Proof: Since R preserves distances, |Ru|= |u| for every u. Let u,v be arbitrary vectorsin X

|u+ v|2 = |u|2 + |v|2 +2Re(u,v)

|Ru+Rv|2 = |Ru|2 + |Rv|2 +2Re(Ru,Rv)

= |u|2 + |v|2 +2Re(R∗Ru,v)

Thus Re(R∗Ru−u,v) = 0 for all v and so by Proposition 12.1.5, (R∗Ru−u,v) = 0 for allv and so R∗Ru = u for all u which implies R∗R = I. ■

The next theorem is interesting. You have a p dimensional subspace of Fn where F= Ror C. Of course this might be “slanted”. However, there is a linear transformation Q whichpreserves distances which maps this subspace to Fp.

Theorem 12.3.6 Suppose V is a subspace of Fn having dimension p≤ n. Then there existsa Q ∈L (Fn,Fn) such that

QV ⊆ span(e1, · · · ,ep)

and |Qx|= |x| for all x. AlsoQ∗Q = QQ∗ = I.

Proof: By Lemma 12.1.6 there exists an orthonormal basis for V,{vi}pi=1 . By using the

Gram Schmidt process this may be extended to an orthonormal basis of the whole spaceFn, {

v1, · · · ,vp,vp+1, · · · ,vn}.

Now define Q ∈L (Fn,Fn) by Q(vi)≡ ei and extend linearly. If ∑ni=1 xivi is an arbitrary

element of Fn, ∣∣∣∣∣Q(

n

∑i=1

xivi

)∣∣∣∣∣2

=

∣∣∣∣∣ n

∑i=1

xiei

∣∣∣∣∣2

=n

∑i=1|xi|2 =

∣∣∣∣∣ n

∑i=1

xivi

∣∣∣∣∣2

.