Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES(b) find an orthonormal basis, within a four-dimensional Euclidean space, forthe subspace spanned by the three vectors (1 2 0 0) T ,(3 − 1 2 0) Tand (0 0 2 1) T .8.14 If a unitary matrix U is written as A + iB, whereA and B are Hermitian withnon-degenerate eigenvalues, show the following:(a) A and B commute;(b) A 2 + B 2 = I;(c) The eigenvectors of A are also eigenvectors of B;(d) The eigenvalues of U have unit modulus (as is necessary for any unitarymatrix).8.15 Determine which of the matrices below are mutually commuting, and, for thosethat are, demonstrate that they have a complete set of eigenvectors in common:( ) ( )6 −21 8A =, B =,C =−2 9(−9 −10−10 5), D =8 −11(14 22 118.16 Find the eigenvalues and a set of eigenvectors of the matrix⎛⎝ 3 4 −21 3 −1⎞⎠ .−1 −2 2Verify that its eigenvectors are mutually orthogonal.8.17 Find three real orthogonal column matrices, each of which is a simultaneouseigenvector ofA =⎛⎝ 0 0 10 1 01 0 0⎞⎛⎠ and B =).⎝ 0 1 11 0 11 1 08.18 Use the results of the first worked example in section 8.14 to evaluate, withoutrepeated matrix multiplication, the expression A 6 x,wherex =(2 4 − 1) T andA is the matrix given in the example.8.19 Given that A is a real symmetric matrix with normalised eigenvectors e i ,obtainthe coefficients α i involved when column matrix x, which is the solution ofAx − µx = v, (∗)is expanded as x = ∑ i α ie i .Hereµ is a given constant and v is a given columnmatrix.⎞⎠ .(a) Solve (∗) whenA =⎛⎝ 2 1 1 2 00⎞⎠ ,0 0 3µ =2andv = (1 2 3) T .(b) Would (∗) have a solution if µ = 1 and (i) v = (1 2 3) T , (ii) v =(2 2 3) T ?310