2. Prove that if the matrices A and B are related by a similarity transformation, A-P-BP,...
1. Prove that the matrices are hermitian or anti-hermitian. 2. Prove that in order for the Dirac equation to be covariant (i.e., form-invariant) under the Lorentz group of trans formations with the constraint that the gamma-matrices are unchanged, the following relation must hold: , where is the Lorentz-transformation matrix for the 4- vectors (x'=Lx) and S is a unitary matrix that transforms the spinor as
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2....
The Hamiltonian of a perturbed 2-dimensional oscillator is given by . Prove that can be diagonalized by the unitary transformation , where is a parameter. Find also the values of such that the transformation is well-defined.
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for the nullspace (Kernel) of T. c) Find a basis for the range of T. 7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for...
QUESTION 6 a) Prove the product of 2 2 x 2 symmetric matrices A and B is a symmetric matrix if and only if AB=BA. b) Prove the product of 2 nx n symmetric matrices A and B is a symmetric matrix if and only if AB=BA.
Please answer both questions.. thank you! :) 5 4 2 1. Give A 4 5 2 (1) Numerically prove that A has an orthonormal basis of eigenvectors. (10%) (2) Find A5 by stating a proper similarity transformation (13%) 5 4 2 1. Give A 4 5 2 (1) Numerically prove that A has an orthonormal basis of eigenvectors. (10%) (2) Find A5 by stating a proper similarity transformation (13%)
HW21 linear transformations transition matrices: Problem 4 Previous Problem Problem List Next Problem 1 point) Recall that similarity of matrices is an equivalence relation, that is, the relation is reflexive, symmetric and transitive. 1 -2 is similar to itself by finding a T such that A TAT Verify that A T= 0 We know that A and are similar since A P-1BP where P Verify that B~A by finding an S such that B- S-'AS Verity that AC by finding...
Extra HW 1. Prove the following properties of the density matrix. (a) ? is a Hermitian operator, i.e. ?-? (b) (A)) is invariant under unitary transformation. (c) Quantum Liouville's equation ih Ot (d) For pure states ?-? and for mixed states ?2 < p.
2.4-13 Suppose A, b, c] is minimal and a(s)det (sI - A) has a repeated root. Prove that A cannot be diagonalzed by a similarity transformation. 2.4-13 Suppose A, b, c] is minimal and a(s)det (sI - A) has a repeated root. Prove that A cannot be diagonalzed by a similarity transformation.
Help! Let B and C be similar nxn matrices. Prove that the matrices given by: I +5B - 2B4 and I +5C - 204 are similar. (6 pts)