2

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?

To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can the eigenvalues/eigenvectors of QRQ be represented by V, D & Q?

Thanks a lot!

Some guys have given some hints for this question on mathoverflow, which I think are helpful. Thank you all for your attention!

Community
  • 1
dehiker
  • 121

1 Answers1

0

let take following example

A=[2 2+i 4;2-i 3 i;4 -i 1]

A =

   2.0000 + 0.0000i   2.0000 + 1.0000i   4.0000 + 0.0000i
   2.0000 - 1.0000i   3.0000 + 0.0000i   0.0000 + 1.0000i
   4.0000 + 0.0000i   0.0000 - 1.0000i   1.0000 + 0.0000i

it's transpose let us check

A'

ans =

   2.0000 + 0.0000i   2.0000 + 1.0000i   4.0000 + 0.0000i
   2.0000 - 1.0000i   3.0000 + 0.0000i   0.0000 + 1.0000i
   4.0000 + 0.0000i   0.0000 - 1.0000i   1.0000 + 0.0000i

now let us take some diagonal element

 X=[2 1 3];

>> y=diag(X)

y =

     2     0     0
     0     1     0
     0     0     3

now about eigenvalue of A

[U D]=eig(A)

U =

   0.6558 - 0.0511i  -0.1537 + 0.1936i  -0.7022 - 0.1147i
  -0.2045 + 0.2342i   0.7745 - 0.3047i  -0.4586 - 0.0179i
  -0.6860 + 0.0000i  -0.4962 + 0.0000i  -0.5321 + 0.0000i


D =

   -3.1651         0         0
         0    2.8531         0
         0         0    6.3120

and 

S=y*A*y

S =

   8.0000 + 0.0000i   4.0000 + 2.0000i  24.0000 + 0.0000i
   4.0000 - 2.0000i   3.0000 + 0.0000i   0.0000 + 3.0000i
  24.0000 + 0.0000i   0.0000 - 3.0000i   9.0000 + 0.0000i

eigenvalue decomposition of $S$

[U1 D1]=eig(S)

U1 =

   0.6992 - 0.0176i   0.1079 + 0.1024i   0.6989 + 0.0118i
  -0.1411 + 0.1789i   0.8189 - 0.5175i   0.0946 + 0.0260i
  -0.6774 + 0.0000i  -0.1986 + 0.0000i   0.7083 + 0.0000i


D1 =

  -16.5679         0         0
         0    3.7771         0
         0         0   32.7908

maybe i am wrong but there should not be any special relation between eigenvalues and eigenvectors

dato datuashvili
  • 9,194