17 Self-Adjoint and Normal Operators 17.1 Adjoints The matrix of T ∗ Let T 2 L(V; W). Suppose B = fe1; :::; eng is an orthonormal basis of V and ∗ 0 Adjoint, T Suppose T 2 L(V; W). The adjoint B = ff1; :::; fmg is an orthonormal basis of W. of T is the function T ∗ : W!V such that Then ∗ [T ]B0B hT v; wi = hv; T ∗wi is the conjugate transpose of for every v 2 V and every w 2 W. [T ]BB0 : 3 2 1. Define T : R ! R by T (x1; x2; x3) = (x2 + 3x3; 2x1) 17.2 Self-Adjoint Operators Find a formula for T ∗. Self-Adjoint (or Hermitian) Operator An operator T 2 L(V) is called self-adjoint if 2. Fix u 2 V and x 2 W. Define T 2 L(V; W) by T = T ∗. In other words, T 2 L(V) is self-adjoint if and only if T v = hv; uix hT v; wi = hv; T wi for all v; w 2 V. ∗ for every v 2 V. Find a formula for T . 2 3. Suppose T is the operator on F whose matrix The adjoint is a linear map (with respect to the standard basis) is mmIf T 2 L(V; W), then T ∗ 2 L(W; V). 2 b : Properties of the adjoint 3 7 (a)( S + T )∗ = S∗ + T ∗ for all S; T 2 L(V; W). Find all numbers b such that T is self-adjoint. ∗ ∗ (b)( λT ) = λT for all λ 2 F and T 2 L(V; W). Eigenvalues of self-adjoint operators are real ∗ ∗ (c)( T ) = T for all T 2 L(V; W). Every eigenvalue of a self-adjoint operator is real. ∗ (d) I = I, where I is the identity operator on Over C, T v is orthogonal to v for all v only V. for the 0 operator mmSuppose V is a complex inner product space (e)( ST )∗ = T ∗S∗ for all T 2 L(V; W) and and T 2 L(V). Suppose S 2 L(W; U) (here U is an inner product space over F). hT v; vi = 0 Null space and range of T ∗ for all v 2 V. Then T = 0. mmSuppose T 2 L(V; W). Then Over C, hT v; vi is real for all v only for self-adjoint operators (a) kerT ∗ = (imT )?. mmSuppose V is a complex inner product space (b) imT ∗ = (kerT )?. and T 2 L(V). Then T is self-adjoint if and only if ∗ ? (c) kerT = (imT ) . hT v; vi 2 R ∗ ? (d) imT = (kerT ) . for every v 2 V. If T = T ∗ and hT v; vi = 0 for all v, then T = 0 Conjugate transpose The conjugate trans- Suppose T is a self-adjoint operator on V such that pose of an m-by-n matrix is the n-by-m matrix obtained by interchanging the rows and columns hT v; vi = 0 and then taking the complex conjugate of each en- try. for all v 2 V. Then T = 0. 27 17.3 Normal operators (b) T is surjective if and only if T ∗ is injective. Normal Operator 9. Prove that • An operator on an inner product space is dim kerT ∗ = dim kerT + dimW − dimV called normal if it commutes with its ad- joint. and dim imT = dim imT • In other words, T 2 L(V) is normal if for every T 2 L(V; W ). TT ∗ = T ∗T: 10. Make P2(R) into an inner product space by defining Let T be the operator on 2 whose matrix 1 4. F hp; qi = p(x)q(x) dx: (with respect to the standard basis) is ˆ0 Define T 2 L(P ( )) by T (a + a x + a x2) = a x. 2 −3 2 R 0 1 2 1 : 3 2 (a) Show that T is not self-adjoint. Show that T is not self-adjoint and that T is normal. (b) The matrix of T with respect to the basis f1; x; x2g is T is normal if and only if kT vk = kT ∗vk for 00 0 01 all v 0 1 0 : An operator T 2 L(V) is normal if and only if @ A 0 0 0 kT vk = kT ∗vk This matrix equals its conjugate transpose, even though T is not self-adjoint. Explain why for all v 2 V. this is not a contradiction. For T normal, T and T ∗ have the same eigenvectors 11. Suppose S; T 2 L(V ) are self-adjoint. Prove Suppose T 2 L(V) is normal and v 2 V is an eigen- that ST is self-adjoint if and only if ST = TS. vector of T with eigenvalue λ. Then v is also an 12. Suppose V is a real inner product space. ∗ eigenvector of T with eigenvalue λ. Show that the set of self-adjoint operators on V is a Orthogonal eigenvectors for normal subspace of L(V ). operators 13. Suppose V is a complex inner product space Suppose T 2 L(V) is normal. Then eigenvectors with V 6= f0g. Show that the set of self-adjoint of T corresponding to distinct eigenvalues are operators on V is not a subspace of L(V ). orthogonal. 14. Suppose dimV ≥ 2. Show that the set of 17.4 Exercises normal operators on V is not a subspace of L(V ). 2 5. Suppose n is a positive integer. Define 15. Suppose P 2 L(V ) is such that P = P . n Prove that there is a subspace U of V such that T 2 L(F ) by P = PU if and only if P is self-adjoint. T (z ; :::; z ) = (0; z ; :::; z ): 1 n 1 n−1 16. Suppose that T is a normal operator on V ∗ and that 3 and 4 are eigenvalues of T . Prove that Find a formula for T (z1; :::; zn). p there exists a vector v 2 V such that kvk = 2 and 6. Suppose T 2 L(V) and λ 2 F. Prove that λ is kT vk = 5. an eigenvalue of T if and only if λ is an eigenvalue Give an example of an operator T 2 L( 4) of T ∗. 17. C such that T is normal but not self-adjoint. 7. Suppose T 2 L(V) and U is a subspace of V. Suppose T is a normal operator on V . Prove that U is invariant under T if and only if U ? 18. Suppose also that v; w 2 V satisfy the equations is invariant under T ∗. 8. Suppose T 2 L(V; W). Prove that kvk = kwk = 2; T v = 3; T w = 4w: (a) T is injective if and only if T ∗ is surjective; Show that kT (v + w)k = 10. 28.