Definition of adjoint

Transcription

1 Definition of adjoint Aim lecture: We generalise the adjoint of complex matrices to linear maps between fin dim inner product spaces. In this lecture, we let F = R or C. Let V, W be inner product spaces with inner products denoted ( ) V, ( ) W. Let D V : V V, D W : W W be the canonical maps. Prop-Defn Let T : V W, T : W V be linear. 1 T T D W = D V T : W V iff for all v, V, w W we have (T w v) V = (w T v) W. 2 Given T, there is at most one linear map T : W V satisfying the condn in 1) above. In this case we write T = T & call it the adjoint of T. 3 If V is fin dim then T exists & T = D 1 V T T D W. Rem The defn of the adjoint depends critically on the inner products involved though the notation does not record this! Proof. For 3), just note that D V is invertible & D 1 V T T D W is linear being the composite of two conjugate linear maps with a linear one. Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

2 Proof cont d For 1) just note T T D W = D V T for all w W, T T (D W w) = D V (T w) for all w W, (D W w) T = (T w ) for all w W, (w T ( )) = (T w ) For 2), suppose T, T both satisfy the condn in 1). Then D V T = D V T. Suffice show for any w W that T w = T w. But D V (T w) = D V (T w) so the result follows from injectivity of D V. Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

3 Fancy example (not examinable) E.g. Let V = Cc be the R-space of compactly supported (= zero outside of some compact set) infinitely differentiable functions on R. Note T = d dt is a lin map from V V. Note that we have an inner product (f g) = f (t)g(t)dt. What s T? A Given f, g V note that integration by parts gives Hence T = d dt. (f dg dt ) = f (t) dg(t) dt dt = [f (t)g(t)] = = ( df dt g) df (t) g(t)dt dt df (t) g(t)dt dt Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

4 Connection with matrix adjoint Q Let A M mn (F) & T A : F n F m be the assoc lin map. What is T A : Fm F n? Answer T A = T A Proof. It suffices to show that for any v F n, w F m we have (A w v) = (w Av). Indeed Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

5 Conjugate linearity & functoriality Prop The following formulae hold whenever they make sense. S, T are appropriate linear maps. 1 (S + T ) = S + T. 2 (βs) = βs for β F. 3 (S T ) = T S. 4 (T ) = T. 5 id = id. In particular, if V, W are fin dim, then the adjoint operator ( ) : L(V, W ) L(W, V ) : S S is conjugate linear. Proof. These all follow from defns & corresponding results for the transpose. For example, Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

6 Orthogonal complements to kernels Prop Let T : V W be a linear map between fin dim inner product spaces. Then 1 (ker T ) = im T 2 If V, W are fin dim then, rankt = rankt. Proof. 1) We show equiv that (im T ) = ker T. Let v V v (im T ) (T w v) = 0 for all w W (w T v) = 0 for all w W 0 = (T v ) = D W (T v) T v = 0, (recall D W is injective) v ker T For 2) we use rank-nullity rankt = dim(im T ) = dim((ker T ) )) = dim V dim(ker T ) = dim(im T ) = rankt. Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7

7 Kernel of T T Prop Let T : V W be linear. 1 If S : W X is linear then ker(s T ) ker T. 2 ker T T = ker T. 3 If V, W are fin dim then rankt T = rankt. Proof. 1) is easy ex. 2) By 1), it suffices to show ker T T ker T so let v ker T T. Then so T v = 0 & v ker T. Part 2) follows. 0 = (T (T v) v) = (T v T v) 3) This follows from the rank-nullity thm since part 2) ensures the nullities are the same (whilst the domains are also the same). Daniel Chan (UNSW) Lecture 36: Adjoints Semester / 7