Question

Given that A= [0 1+2i ,-1+2i 0] show that
(1-A)(1+A)^(-1) is a unitary Matrix.​

Answer 1

Answer:

1. Find the eigenvalues and corresponding eigenvectors of matrices(a)[1  14  1](b)−1   2   22   2   2−3−6−6Solution:(a) (1−λ)2−4 = 0⇒(λ−3)(λ+ 1) = 0⇒λ1= 3,  λ2=−1. Also,v1= [ 1/2 1 ]T,  v2=[−1/2 1 ]T.(b)λ1= 0, λ2=−2, λ3=−3 andv1= [ 0−1 1 ]T, v2= [−2 1 0 ]T, v3= [−1 0 1 ]T.2. Construct a basis ofR3consisting of eigenvectors of the following matrices(a)0  0  20  2  02  0  3(b)1  1−1−1  1   1−1  1   1.Solution:(a) Eigenvalues areλ1=−1, λ2= 2, λ3= 4 and eigenvectors arev1= [ 1 0−1/2 ]T,v2=[ 0 1 0 ]T,v3= [ 1 0 2 ]T. Since eigenvectors corresponding to different eigenvaluesarelinearly independent,{v1,v2,v3}is a basis ofR3.(b) Similar to (a).3.(T)This question deals with the following symmetric matrixA:A=1  0  10  1  -11  -1  0.One eigenvalue isλ= 1 with the line of eigenvectorsx= (c, c,0).(a) That line is the null space of what matrix constructed fromA?Solution:The eigenvectors ofλ= 1 makes the null space ofA−I.(b) Find the other two eigenvalues ofAand two corresponding eigenvectors.Solution:Ahas trace 2 and determinant−2. So the two eigenvalues afterλ1= 1 will addto 1 and multiply to−2. Those areλ2= 2 andλ3=−1. Corresponding eigenvectors are :v2=1-11,   v3=1-1-2.

2(c) The diagonalizationA=SΛS−1has a specially nice form becauseA=At. Write all entriesin the three matrices in the nice symmetric diagonalizationofA.Solution:Every symmetric matrix has the nice formA=QΛQtwith an orthogonal matrixQ. The columns ofQare orthonormal eigenvectors.Q=1/√2   1/√3   1/√61/√2−1/√3−1/√601/√3−2/√6Λ =1  0  00  2  00  0  

Step-by-step explanation: