5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm
5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm
6.2.3 Let U be a complex vector space with a positive definite scalar product and S, T e L(U) self-adjoint and commutative, so T-T o S. (i) Prove the identity 11(S iT)(u)ll-llS(11 )11 2 + llT(11)112, 11 e U. (6.2.10) (ii) Show that S ± iT is invertible if either S or T is so. However, the converse is not true. (This is an extended version of Exercise 4.3.4.)
6.2.3 Let U be a complex vector space with a positive...
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
1. Let 1 -1][-1 s={ 112 [1] 1 1 Find a basis for the subspace W = span S of M22. What is the dim W? 2. Find the basis for the solution space of the homogeneous system: a. x+2y = 0 2x+4y =0 b. 3x+2y+4z=0 2x+ y - Z = 0 x +y +3z =0
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix.
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
Topology
(b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that for every r E co and every ε > 0, there exists y S such that llx-yI100 < ε.j (ii) Conclude that (co, ll . 114) is separable (only quote relevant results) (iii) Show that the closed unit ball in (a-II ·...
4.5.17 Let V denote R^2 equipped with the l^ 1 norm and let W denote R^2 equipped with the l^∞ norm. Show that the linear map in L(V,W) represented by [ 1 1 ] is an isometry. Remark: On the other hand, if n ≥ 3, then R^n with the l^1 norm and R^n with the l^∞ norm are not isometric. [ 1 -1 ]
(Functional analysis) Let C be the space of all functions
having
Question 4. (3 marks) Let C([0, 1]) be the space of all functions having continuous derivative For each fe C(0,1), set 1/2 1 0 Show that I-1l is a norm of the space of C (0, 1)
Question 4. (3 marks) Let C([0, 1]) be the space of all functions having continuous derivative For each fe C(0,1), set 1/2 1 0 Show that I-1l is a norm of the...
(5) Let S denote the set of sequences whose series are absolutely convergent. We define two norms on S by ll{an} olla ΣΙΑ Janlı {an) oloup = sup{lano- no 1 (Note that S is the set of sequences such that a1 <. The sup-norm is sometimes called the o-norm.) Define a linear operator : S R by E({en}_o) - an NO (i) Compute the operator norm of using |-|- (ii) Show that the operator norm of using sup is unbounded.