15. (i) Let A, B be 2 × 2 matrices over R and vectors x, y in R2such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that ABand BA have an eigenvalue +1.„T,%3D(ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Usex =sin(a) )– sin(a) )cos(a)y =

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Answered: 15. (i) Let A, B be 2 x 2 matrices over…

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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Write the given second order equation as its equivalent system of first order equations. u" + 5u' + 7u = 0 Use v to represent the "velocity function", i.e. v = Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u'(t). u' = Now write the system using matrices: u d dt V
The eigenvalues of the system x' = x y =x are a) {i,-i} b) {0,1} c) {-1,0} d) {-1, 1}
Only (a) and (d) and (e).
Consider B={1,x,x^2} and B'={1+x,1-x^2,2} Find the change of basis matrix 1.P from Bto B' 2.Q from B' to B Further find the relation between P and Q
For the system of differential equations, X1, X2 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U2 = = y' = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue A2. 5 2 - 1 Y2 (t) d) Determine two linearly independent solutions to the system. Enter the first solution in the format y₁ (t) y = = y₁ (t) Enter the second solution in the format y₂(t) = ƒ2(t) (v3h₁(t) + v₁h₂(t)) = : fi(t) (vigi(t) – v2g2(t)) . +
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
2 [H] a) Treating u, v', and w' as vectors, are the following inner products defined? (i) u. v' 6) (True/False) Let u = (ii) v'. u (iii) u - w' v' = [1 3 2], and w' = [5 -1 -1]. b) Treating u, v', and w' as matrices, are the following matrix products defined? (i) uv' (ii) v'u (iii) uw'
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15. (i) Let A, B be 2 x 2 matrices over R and vectors x, y in R2 such that Ax = y, By = x, x"y = 0 and x"x = 1, y"y = 1. Show that AB and BA have an eigenvalue +1. %3D %3D (ii) Find all 2 × 2 matrices A, B which satisfy the conditions given in (i). Use cos(a) x = - sin(a)) cos(a) y = sin(o)),