4. For a unitary operator U, show that T = -ilog(U) is Hermitian and therefore thatU = eil for some Hermitian T.

Answered: -i log(U) is Hermitian and therefore…

Math

Advanced Math

-i log(U) is Hermitian and therefore that 4. For a unitary operator U, show that T = U = el for some Hermitian T.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

See similar textbooks

Related questions

Q: A. Expand, that is, perform the indicated operations on the differential operator D. 1. xD(D + x) 2.…

Q: 0 0 0 1 0 0 0 1 0 ) Let A =

Q: Determine the Fourier transform of the following: f(t) = {e¹³t 0, −1<t<1 elsewhere

Q: Find the fourier transform of the following. functions by making use of Jordan's Lemma. a) 1 (x² +…

Q: Calculate the Laplace transform of the following: f(t)=cos^2(3t) 1 S а. 2s2+72 3s b. 2s²+71 1 с. -…

A: Use the formula: cos2t=cos 2t +12

Q: (a) Find if y = y (ln r). (b) The function f(z) (In r) is an invertible function (you do not need to…

Q: If the Fourier transform of u(x, t)=2 e 2"w then u(x, t) = 812 2t TT O b. 1. e 812 x² e Od. 2 412 e…

A: Given Fourier transform of ux,t is Fux,t=2e-2t2w2 To find ux,t

Q: Consider the function f : R → R defined by J 1, 1, if 0≤ x ≤1 f(x) = { 0 else Compute explicitly its…

Q: ) Compute the Fourier transform of the function. (a) f(x) = h(x + 1) – 2h(x) + h(x - 1), where 1, if…

Q: 3. Find the Fourier transform of the following: f(t) = {te-t -1<t<0 elsewhere

A: We have to find the Fourier transform of the given function.

Q: ) Determine the complex exponential Fourier transform for the following periodic signal shown. x(t)…

A: To find : Complex exponential Fourier transform for given signal

Q: Let z be defined implicitly as a function of x and y by the e e-zy əz find O UTM + cos (xy) = (4 –…

Q: Consider a function f(t) with a finite discontinuity at t = 0. Given that its Fourier transform is…

Q: 15. The DI diagram to help solve the Fourier coefficient bn tsin given that %3D f(t) = f(t + 4) is:

Q: Determine continuous time Fourier transform for the given signals. x(t) 1 1, t20 0, t < 0

Q: 8. Prove that the differential operator M: C² [a,b] → C[a,b] defined by d²u du Mu= 2+3u dx is a…

Q: 2) Find where tan'z = Logit is analytic a) C\{Re(z)=0, Im(z) (-0,1]} b) C\{Re(z)=0, Im(z) (-00,1] U…

Q: 1. Determine whether the functions f1(x) = cos 3x, f2(x) = x, and f3(x) = cos² x are linearly…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

Q: -21 e * u(t) + e ª u(1)

A: The given function is e-tut+e-2tut Let xt=e-tut and x1t= e-2tut

Q: Suppose w=x/y+y/z, where x=e3t, y=2+sin(t)x=e3t, and z=2+cos(6t). A ) Use the chain rule to find…

A: w=xy+yz, where x=e3t, y=2+sint & z=2+cos6t Partially differentiating w=xy+yz with respect to x,…

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

Q: Show that the matrix cos e sin e 01 A = =- sin e cos e 1 is invertible for all values of e; then…

A: To show that the matrix is invertible.

Q: Find the Fourier Transform of Hence Prove that f(x): cos(sx) + Ssin(sx) 1+s² -ds = x 0 OEIN 0 2 Tex…

A: Introduction: Through Fourier transformation, a real-time function can be converted into a complex…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

Q: L 2 + 1 s(s − 1)(s + 1)(s − 4),

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

Q: L 4s 2 4s² + 1

Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D

A: As it's mentioned in the question to answer between a) or b), so showing steps for only one part. b)…

Q: Show that the function f:R –R givenby f(x) = log, (x+vx* +1),a>o,a±1. is invertible and find its…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

A: We will solve this question by Laplace transform

Q: . Let Show that, Hence show that •2T F₁(x) = 2/1/² 0 Fn(2) = eix cos ∞ N=-∞ eix cose eine de in Jn…

A: As per the question we are given the sequence of functions Fn(x) in form of a Fourier integral. And…

Q: 2.5s (s 0.1) (s+ 0.4),

Q: (B) Compute the discrete Fourier transform of g = t+1 on [0, 4) with length 3. F{8} = (C

A: To calculate the discrete Fourier transform of the below function on [0,4) with length 3. g=t+1

Q: Prove that the function f = (sin kæ)(sin my)(sin nz) where: k, m, n are constants 82 + dx² ' Sy? '…

A: f(x) is an eigen function of an operator F(D) if F(D)f(x)=kf(x), where k is a constant, known…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

Q: Let T: C* (R) → C (R), T(f) = f' – 2f be a (linear) differential operator on the space of infinitely…

Q: Prove that the Generating function of Bessel function i.e. £1J„(x)t" = e*/2(-1/2), %3D

A: Let ϕx,t=ex2t-1t. Let us prove that ϕx,t is the generating function of the Bessel function…

Q: Find the autocorrelation function, R(t) of the following Gaussian function: z1) = (). exp 2TO 202…

A: # auto correction function of Gaussian function X(t) is =1/(2*sqrt(π)*sigma)* e^-T^2/4sigma^2

Q: 3. Determine the Fourier transform of the following: f(t) = {el³6 0, -1 < t < 1 elsewhere

A: Fourier transform of f(t) F(f(t))=∫-∞∞f(t).ejstdt given function f(t)=ej3t-1<t<10elsewhere

Q: Find the Wronskian for y, = sin 3x , and y2 = cos 3x.

A: The Wronskian, W, of two functions, f and g is calculated as follows. W=fgf'g' , where, f' , g'are…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

A: Given, L-11.4ss-0.1s+0.6

Q: Suppose w=x/y+y/z, where x=e^(4t), y=2+sin⁡(4t), and z=2+cos⁡(3t). A ) Use the chain rule to find…

Q: 13. Use the chain rule, in Leibniz notation, to find dx dy at the given value of x. а. у %3D Зи? —…

A: We are solving the first three subparts and for the next subparts post the question againand also…

Q: Solve the following problem using Fourier transform %3D (oave J Utt +4 uxx =0 Claplan) xE (-00, 00),…

A: We will use basic knowledge of Fourier transform and inverse Fourier transform to answer this…

Q: A. Compute the Fourier transform X(jw) for the following aperiodic signal. Also show the spectrum as…

A: We have to find the Fourier transform of the given aperiodic signal. The function x(t) is…

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

A: Laplace transform is a transformation that convers the real variable into a complex variable. To…

Question

4. For a unitary operator U, show that T = -ilog(U) is Hermitian and therefore that
U = eil for some Hermitian T.

Expert Solution

Step 1

Let U be a unitary operator.

To prove that $T = - i \log (U)$ is Hermitian.

Since, U is an unitary operator, $U U^{*} = U^{*} U = I$ where $U^{*}$ is the adjoint of U and $I$ is the identity operator.

As $T = - i \log (U)$ , therefore, it's adjoint is $T^{*} = i \log (U^{*})$

Now,

$\begin{array}{rcl} T - T^{*} & = & - i \log (u) - [i \log (U^{*})] \\ = & - i \log (U) - i \log (U^{*}) \\ = & - i \log (U) - i \log (U^{- 1}) \\ = & i \log (U) + i \log (U) [\log (A^{- 1}) = - \log (A)] \\ = & 0 \\ T & = & T^{*} \end{array}$

Hence, T is Hermitian.

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Matrix Eigenvalues and Eigenvectors$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
Must show that when A is invertible, x, y, z, w are distinct AND when x, y, z, w are distinct, A is invertible
For a linear operator T on an inner product space V, prove that T∗T = T0 implies T = T0. Is the same result true if we assume that TT∗= T0?
Define the operators Δ, (, E, μ and δ.Prove the relation μ= √(1+¼ δ²) and μ-1= 1 – δ2/8+3 δ4 /128.Prove the relation μ=cosh((u/2) where u=hD, D is differential operator.
How can I prove that in terms of this inner product the operator ∂t : s(t) → ∂ts(t) is anti-Hermitian.
Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then BTB is positive definite.
For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint, or unitary.
Prove that if A is invertible and BA = CA, then B = C.Give a counterexample to show that the result may fail if A is not invertible.