18.03(4) at ESG Spring, 2003Practice ProblemsI - Properties of Hermitian MatricesFor scalars we often consider the complex conjugate, denotedz in our notation.For matrices, we often consider the Hermitian Conjugate of a matrix, which is thetranspose of the ma trix of complex conjugates, and will be denoted by A†(it’s aphysics thing). Specifically, if the entries of A are ajk, the entries of A†areakj.If A is an m × n matrix (m rows, n columns), A†is an n × m matrix. In longer,more explicit form,if A =a11a12· · · a1na21a22· · · a2n.........· · ·am1am2· · · amn, A†=a11a21· · · am1a12a22· · · am2.........· · ·a1na2n· · · amn.A square matrix is Hermitian if A = A†and Skew-Hermitian if A = −A†.(a) To see how this works in a simple case, let x ∈ Cmbe a column vector(an m × 1 matrix) of complex elements xj, j = 1 . . . m, and define t he square of thenorm of x to be|x |2=x1x1+ x2x2+ · · · xmxm=mXj=1xjxj.Show that this can be expressed as|x |2= x†x .The rest of this practice problem will consider only 2 × 2 matri x o perators and2 × 1 column vectors and their 1 × 2 Hermitian cojugates. If a 2 × 2 matrix isHermitian, we’ll denote this matrix asH =acc b,1where a and b are real and c is complex (real, imaginary or neither). Note that wecould have put the overline representing scalar complex conjugation in the lowerleft instead of the upper right.(b) Show that the eigenvalues of H are real. Do t his two ways, first bydirect computation of the roots of p( λ) = det (H − λ I) and then by considering theproduct f†H f for f an eigenvector of H. The point of doing this two ways is thatthe first method sort of craps out for larger matrices, while the second is valid forHermitian matrices of any size.(c) Show that if the eigenvalues of H are equal, then H is a real scalar multipleof I, a nd hence not worth special consideration (every vector an eigenvector). Showthat if the eigenva lues are not equal, then the eigenvectors are ort hogonal; in thiscontext, two vectors x and y are orthogonal if x†y = 0 (this is consistent withpart (a) above.) Again, do this two ways, first by direct computation (there a re afew tricks to look for) and then by considering the product f2†H f1where f1and f2are eigenvectors corresponding to different eigenvalues. As in (b) above, the secondmethod is valid for Hermitian matrices of any size.(d) Show that the sum of any two Hermitian matrices is Hermitian (if they arethe same size, of course), H2is Hermitian, but for two arbitrary Hermitian matrices(if they are the same size, of course), the product is not necessarily Hermitian.Speculate about H3and higher powers, invoking the Cayley-Hamilto n Theorem (ordemonstrating the validity of the theorem directly).(e) Yo u knew this was coming: Find exp (H t).In doing parts (b) and (c) using direct computation, you’ve found that t he generalform for the eigenvalues and eigenvectors is a bit of a mess and diagonalizationwould be unpleasant, so here’s what we’ll do. ExpressH =a + b2I +dcc −dwhere d = (a − b)/2. The first matrix on the right above commutes with the secondmatrix (I’m tempted to say “clearly”, since the first is a scalar multiple of theidentity, but I won’t), soexp (H t) = e(a+b)t/2expdcc −dt.2Finddcc −d2, express your delight in a seemly manner, and go from there. A sug-gestion is to simplify t he form by introducing the parameter ω2= d2+c c.II - Manipulating Solutions(A) It is known that two functions y1and y2satisfy a linear differential equa-tion of the formy′′+ P (t) y′+ Q(t) y = 0.Given y1and y2, can you determine P (t) and Q(t), and if so, what are they in termsof y1, y2, and their derivatives?(B) Find, if possible, P and Q when y1= et, y2= t et.(C) Find, if possible, P and Q when y1= cos(2t), y2= 1 − 2 cos2t.III - Differential Operators as MatricesConsider the set of scalar functionsx1= cos2t, x2= sin2t, x3= sin 2t.Let yj=dxjdt, j = 1, 2, 3, and denotey =y1y2y3, x =x1x2x3.Find a constant matrix D such that y = D x . Find the rank of D and hence thedimension of the space spanned by the yi, and explain briefly. Find D2and D3,and i dentify the constant scalar c such that D3= c D.IV - Rotation, Rotation, Rotation(a) Find all real 2 × 2 mat rices B such that if x has real components,|B x | = |x |for any x , where |x | is the Euclidean norm. These are the 2 × 2 rotation matrices.(b) Show t hat the matrix that represents T : R2→ R2, Txy=−x−yis arotation matrix. (Note: if you find this really easy, you’re right.)3(c) Find the determinant of an arbit ra ry matrix B of the form you found inpart (a).(d) Describe the rotations in three dimensions represented byG1=0 1 0−1 0 00 0 1, G2=0 0 10 1 0−1 0 0, G3=1 0 00 0 10 −1 0.Pick any two of the above matrices and show that they don’t commute. Findexp (G1t) , exp (G2t) , exp (G3t) .(There are several ways to do this, of course. Hints avai lable on request).(e) Explain why there is no rotation mat rix F such thatFxyz=−x−y−z.If you can’t find the algebra proof, explain in
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