We define a linear bijection, h, between R4 and H(2), the set ofcomplex 2 x 2 hermitian matrices, byy – izt - xt +x(t, z, y, 2) → (,y + izWe denote the matrix on the right-hand side by H.(i) Show that the matrix can be written as a linear combination of the Paulispin matrices 01, 02, 03 and the identity matrix I2.(ii) Find the inverse map.(iii) Calculate the determinant of 2 x 2 hermitian matrix H. Discuss.

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Answered: We define a linear bijection, h,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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d) Transform the system of equations if possible: x'= 1 2 1x. 2 11 into a decoupled system, by means of a similarity transformation, and compute a fundamental matrix of it.
Let A, B be n x n matrices over C. A scalar product can be Problem defined as (A,B) := tr(AB*). = tr(AA*). This norm is The scalar product implies a norm || A||² = (A, A) called the Hilbert-Schmidt norm. (i) Consider the two Dirac matrices 10 0 0 0 0 0 1 0 0 10 70 := 01 0 0 00-1 71 0 0 -1 00 0 0 0 -1 -1 0 00 Calculate (70, 71). (ii) Let U be a unitary n × n matrix. Find (UA,UB). (iii) Let C, D be m x m matrices over C. Find (A C, B D). (iv) Let U be a unitary matrix. Calculate (U,U). Then find the norm implied by the scalar product. (v) Calculate ||U|| := max||v||=1 ||Uv||.
Need only a handwritten solution only (not a typed one).
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3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them
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Let H:R* → R2 be defined by H(x1, x2, X3, X4) = (x3 + x4, X1). Find the standard matrix for H. (H is linear and no need to verify) 9.
Hand written plz Consider the full and reduced singular value decompositions (SVD) of a square matrix A = UΣVH, for both SVDs, which of the flowing statements is correct: [1] U, V must be the same orthogonal matrices; [2] U-¹ = UH, VH = V-¹; [3] Σ must be different from each other; [4] U, V may have the same rank. (a) [1], [2], [3], [4] (b) Only [2] (c) Only [4] (d) None of [1], [2], [3], [4]
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'
1. Let matrix A, B, A + B, A + B are nxn non-singular matrices В, matrix means it has an inverse), prove: B(A + B) + (A' +B)' B = I

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